Celebratio Mathematica

phdthesis M. H. Freedman: Codimension-two surgery. Ph.D. thesis, Princeton University, 1973. Advised by W. Browder. MR 2623579

article M. Freedman: “On the classification of taut submanifolds,” Bull. Amer. Math. Soc. 81 : 6 (1975), pp. 1067–1068. MR 0413143 Zbl 0331.57013

incollection M. H. Freedman: “Automorphisms of circle bundles over surfaces,” pp. 212–214 in Geometric topology (Park City, UT, February 19–22, 1974). Edited by L. C. Glaser and T. B. Rushing. Lecture Notes in Mathematics 438. Springer, 1975. MR 0391140 Zbl 0308.55006

T. Benny Rushing	Related
Leslie C. Glaser	Related

article M. Freedman: “Uniqueness theorems for taut submanifolds,” Pacific J. Math. 62 : 2 (1976), pp. 379–387. MR 0407860 Zbl 0357.57004

article M. H. Freedman and L. Taylor: “\( \Lambda \)-splitting 4-manifolds,” Topology 16 : 2 (1977), pp. 181–184. MR 0442954 Zbl 0363.57004

book M. H. Freedman: Surgery on codimension-2 submanifolds, vol. 12. Memoirs of the American Mathematical Society 191. American Mathematical Society (Providence, RI), September 1977. MR 0454990 Zbl 0369.57012

incollection M. Freedman and R. Kirby: “A geometric proof of Rochlin’s theorem,” pp. 85–97 in Algebraic and geometric topology (Stanford University, CA, August 2–21, 1976), part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics XXXII. American Mathematical Society (Providence, RI), 1978. MR 0520525 Zbl 0392.57018

In 1974 Andrew Casson outlined to us a proof of Rochlin’s Theorem on the index of a smooth, closed 4-manifold \( M^4 \). His proof involved the Arf invariant of a certain quadratic form defined on the first homology group of a surface in \( M^4 \) which is dual to the second Stiefel–Whitney class of \( M^4 \). Our proof was derived from Casson’s; it is the same in principle but differs considerably in detail. After this manuscript was written, we discovered that Rochlin had already in 1971 given a short sketch of this proof; it appears in a paper [Rochlin 1972] about real algebraic curves in \( \mathbb{R}P^2 \).

In addition we obtain a “stable” converse to the Kervaire–Milnor nonimbedding theorem [Kervaire and Milnor 1961], and in Section 2, by relaxing some orientability assumptions, we prove a new (but unspectacular) nonimbedding theorem and find an obstruction to approximating unoriented simplicial 3-chains in a 5-manifold by an immersed 3-manifold.

Richard James Milgram	Related
Robion C. Kirby	Related	Volume

incollection M. H. Freedman: “Remarks on the solution of first degree equations in groups,” pp. 87–93 in Algebraic and geometric topology: Proceedings of a symposium in honor of Raymond L. Wilder (Santa Barbara, CA, July 25–29, 1977). Edited by K. C. Millett. Lecture Notes in Mathematics 664. Springer (Berlin), 1978. MR 0518409 Zbl 0385.20023

Kenneth C. Millett	Related
Raymond Louis Wilder	Related

article M. Freedman and W. H. Meeks, III: “Une obstruction élémentaire à l’existence d’une action continue de groupe dans une variété,” C. R. Acad. Sci. Paris Sér. A 286 : 4 (1978), pp. 195–198. MR 0515866 Zbl 0373.57021

It is natural to think that most manifolds have no symmetry, and therefore do not admit any nontrivial compact Lie group action. However it has been hard to find such manifolds: The first example was constructed by P. E. Conner, F. Raymond and P. Weinberger [1972]. Even the problem of excluding the effective actions of certain compact Lie groups has proved to be difficult [Atiyah and Hirzebruch 1970; Hirzebruch 1975; Yau 1977].

We prove that if a smooth manifold \( M \) admits an effective action of the circle \( \mathbb{S}^1 \), then its de Rham cohomology has certain properties from which one can deduce geometric properties of \( M \) that do not appear on its cohomology alone. For example, if \( M \) is a compact \( n \)-manifold, not homotopy equivalent to the \( n \)-sphere \( \mathbb{S}^n \), then the connected sum \( M\mathbin{\#}\mathbb{T}^n \) of \( M \) with the \( n \)-torus \( \mathbb{T}^n \) admits no effective action of the circle \( \mathbb{S}^1 \).”

article M. H. Freedman: “Quadruple points of 3-manifolds in \( S^{4} \),” Comment. Math. Helv. 53 : 3 (1978), pp. 385–394. MR 0505553 Zbl 0404.57011

A folk theorem (see Banchoff [1974]) says that the number of normally triple points of a closed surface normally immersed in 3-space is congruent modulo two to its Euler characteristic. In general, a normal immersion of a compact \( n \)-manifold in an \( n + 1 \)-manifold will have a finite number, \( \theta \), of \( (n + 1) \)-tuple points. \( \theta \), taken mod 2, is well defined under bordism of both the immersion and ambient manifold. An attractive place to try to evaluate \( \theta \) is on the abelian group, “(oriented bordism of immersed \( n \)-manifolds in \( \mathbb{S}^{n+1} \), connected sum)” \( = B_n \), since \( B_n \) is naturally isomorphic to the stable homotopy group \( \pi_n \). Counting \( (n + 1) \)-tuple points determines a homomorphism, \( \theta_n: \pi_n \rightarrow\mathbb{Z}_2 \). The figure eight immersion of a circle shows that \( \theta_1 \) is an isomorphism; Banchoff’s proof shows that \( \theta_2 \) is the zero map; the main result of this paper is that \( \theta_3 \) is the unique epimorphism \( \pi_3 \simeq\mathbb{Z}_{24} \rightarrow\mathbb{Z}_2 \). Thus, we show that a (actually any) oriented 3-manifold may be generically immersed in \( \mathbb{S}^4 \) with an odd number of quadruple points. Like Smale’s inversion of \( \mathbb{S}^2 \), our proof is abstract and does not yield an example.

article M. H. Freedman: “A fake \( S^{3}\times \mathbf{R} \),” Ann. of Math. (2) 110 : 1 (1979), pp. 177–201. MR 0541336 Zbl 0442.57014

This paper describes a construction which can be exploited to recover the results of higher dimensional surgery (\( \dim \geq 5 \)) for a wide variety of noncompact four dimensional surgery problems. These include any countable union over boundary components of compact problems with simply connected targets. Here we will concentrate on a special case: the construction of a 4-manifold \( W \) proper-homotopy-equivalent to \( \mathbb{S}^3\times\mathbb{R} \) which has a cross section diffeomorphic to the Poincaré sphere \( \Sigma^3 \) and thus cannot be diffeomorphic to \( \mathbb{S}^3\times\mathbb{R} \). Although the ideas of surgery are in the background, our arguments do not rely heavily on them. A sequel will appear to develop a version of the surgery exact sequence for noncompact 4-manifolds.

article M. H. Freedman: “A converse to (Milnor–Kervaire theorem)\( \times R \) etc…,” Pacific J. Math. 82 : 2 (1979), pp. 357–369. MR 0551695 Zbl 0459.57020

article M. H. Freedman: “Cancelling 1-handles and some topological imbeddings,” Pacific J. Math. 80 : 1 (1979), pp. 127–130. MR 0534700 Zbl 0416.57016

In this note we use the existence of a certain type of handle decomposition (see corollary) for compact simply connected P.L. 4-manifolds and R. Edwards’ results on the double suspension conjecture to prove:

Let \( \alpha \in H_2(M;\mathbb{Z}) \) where \( M \) is a compact simply connected P.L. 4-manifold. Then there is a proper topological imbedding (possible nonlocally flat) \[ \theta: \mathbb{S}^2\times \mathbb{R}\rightarrow M\times \mathbb{R} \] (mapping ends to ends) with \[ \theta_*[\mathbb{S}^2\times\mathbb{R}] = \bar{\alpha}\in H_2(M\times\mathbb{R};\mathbb{Z}) .\] \( \bar{\alpha} \) is the image of \( \alpha \) under \( {}\times\mathbb{R} \). Proper, here, means inverse images of compact sets are compact.

incollection L. Siebenmann: “Amorces de la chirurgie en dimension quatre: un \( S^{3}\times \mathbf{R} \) exotique [d’après Andrew J. Casson et Michael H. Freedman],” pp. 183–207 in Séminaire Bourbaki (1978/79). Lecture Notes in Mathematics 770. Springer (Berlin), 1980. Exposé no. 536. MR 572425 Zbl 0444.57021

Andrew J. Casson	Related
Laurent C. Siebenmann	Related

article M. H. Freedman: “Planes triply tangent to curves with nonvanishing torsion,” Topology 19 : 1 (1980), pp. 1–8. MR 0559472 Zbl 0438.53001

article M. Freedman and F. Quinn: “A quick proof of the 4-dimensional stable surgery theorem,” Comment. Math. Helv. 55 : 4 (1980), pp. 668–671. MR 0604722 Zbl 0453.57024

M. H. Freedman: [Unpublished handwritten notes].

article M. Freedman and F. Quinn: “Slightly singular 4-manifolds,” Topology 20 : 2 (1981), pp. 161–173. MR 0605655 Zbl 0459.57008

article M. H. Freedman: “A surgery sequence in dimension four; the relations with knot concordance,” Invent. Math. 68 : 2 (1982), pp. 195–226. MR 0666159 Zbl 0504.57016

We present a systematic treatment of the classification problem for compact smooth 4-manifolds \( M \). It is modeled on the surgery exact sequence, the central theorem in the classification of \( n \)-manifolds \( n \geq 5 \). The price for the extension to dimension \( = 4 \) is a hole in \( M \) where a (homotopy) 1-skeleton should be. There is no homotopy theoretic or surgical obstruction to completing \( M \) with a wedge of circles so that the completion has the topology of a compact smooth manifold. This point-set problem is all that stands between 4-manifolds and the tranquility that prevails in higher dimensions.

When \( M \) is simply connected only a point is missing from the model. The applications of this are discussed in [Freedman 1979] and [Freedman and Quinn 1981]. The general theory has application to knot and link theory. In particular, knots with Alexander polynomial \( = 1 \) are characterized geometrically as knots admitting a certain type of “singular slice.” As a lure to low-dimensional topologists, the knot theoretic “applications” are actually presented first as a special case.

This paper was written in 1979. To bring it up-to-date [Freedman 1982] with recent developments one should say that an isolated simply connected end of a smooth 4-manifold is now known to be topologically collared as \( \mathbb{S}^3\times [0,\infty) \). Thus none of the isolated singularities of 4-manifolds contemplated in this paper actually exists in a topological sense. Also the solution to the 4-dimensional Poincaré conjecture identifies a homotopy \( \mathbb{B}^4 \) with \( \mathbb{S}^3 \) boundary as topologically \( \mathbb{B}^4 \). Thus, for example, the untwisted doubles of a knot with Alexander polynomial \( = 1 \) are actually sliced by topologically flat disks in \( \mathbb{B}^4 \). However, the singular nature of the homotopically flat disks and the nonsimply connected ends which we encounter is still an open question.

article M. Freedman, J. Hass, and P. Scott: “Closed geodesics on surfaces,” Bull. London Math. Soc. 14 : 5 (1982), pp. 385–391. MR 0671777 Zbl 0476.53026

Let \( M^2 \) be a closed Riemannian 2-manifold, and let \( \alpha \) denote a non-trivial element of \( \pi_1(M) \). The set of all loops in \( M \) which represent a has a shortest element \( f:\mathbb{S}^1 \rightarrow M \), which can be assumed smooth and which will be a closed geodesic. (We say a loop represents \( \alpha \) when it represents any conjugate of \( \alpha \). Such a loop need not pass through the base point of \( M \).) The map \( f \) cannot be unique, because \( f \) is not necessarily parametrised by arc length and because there is no base point. In general, even the image set of a shortest loop is not unique. In this note, we prove the following result.

Let \( M^2 \) be a closed, Riemannian 2-manifold and let \( \alpha \) denote a non-trivial element of \( \pi_1M \) which is represented by a two-sided embedded loop \( C \). Then any shortest loop \( f:\mathbb{S}^1 \rightarrow M \) representing \( \alpha \) is either an embedding or a double cover of a one-sided embedded curve \( K \). In the second case, \( C \) bounds a Moebius band in \( M \) and \( K \) is isotopic to the centre of this band.

Joel Hass	Related	Volume
Peter Scott	Related	Volume

article M. H. Freedman: “The topology of four-dimensional manifolds,” J. Differential Geom. 17 : 3 (1982), pp. 357–453. MR 0679066 Zbl 0528.57011

Manifold topology enjoyed a golden age in the late 1950s and 1960s. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained.

L. Siebenmann: “La conjecture de Poincaré topologique en dimension 4 (d’après M. H. Freedman),” pp. 219–248 in Bourbaki Seminar, vol. 1981/1982. Astérisque 92. Soc. Math. France, Paris, 1982. MR 689532 incollection

incollection M. H. Freedman: “A conservative Dehn’s lemma,” pp. 121–130 in Low dimensional topology (San Francisco, CA, January 7–11, 1981). Edited by S. J. Lomonaco, Jr. Contemporary Mathematics 20. American Mathematical Society (Providence, RI), 1983. MR 0718137 Zbl 0525.57005

article M. Freedman, J. Hass, and P. Scott: “Least area incompressible surfaces in 3-manifolds,” Invent. Math. 71 : 3 (1983), pp. 609–642. MR 0695910 Zbl 0482.53045

Let \( M \) be a Riemannian manifold and let \( F \) be a closed surface. A map \( f:F\rightarrow M \) is called least area if the area of \( f \) is less than the area of any homotopic map from \( F \) to \( M \). Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function.

In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly.

Joel Hass	Related	Volume
Peter Scott	Related	Volume

article M. Freedman and S.-T. Yau: “Homotopically trivial symmetries of Haken manifolds are toral,” Topology 22 : 2 (1983), pp. 179–189. MR 0683759 Zbl 0515.57004

M. Freedman and K. Uhlenbeck: Gauge theories and four-manifolds. Technical report 025-83, U.C. Berkeley, 1983. Notes by D. Freed and K. Uhlenbeck.

Karen Uhlenbeck	Related	Volume
Daniel Stuart Freed	Related

incollection M. H. Freedman: “The disk theorem for four-dimensional manifolds,” pp. 647–663 in Proceedings of the International Congress of Mathematicians (August 16–24, 1983, Warsaw), vol. 1. Edited by Z. Ciesielski and C. Olech. PWN (Warsaw), 1984. MR 0804721 Zbl 0577.57003

The two-dimensional disk \( \mathbb{D}^2 \) seems to serve as a fundamental unit in manifold topology, mediating algebra and geometry. For manifolds of dimension greater than or equal to five, intersection pairings taking values in group rings \( \mathbb{Z}[\pi_1M] \) are crucial to the classification problem. The pairings are translated into precise geometric information by isotopies guided by imbedded two-disks. This is the “Whitney trick” [Whitney 1944], key to both \( s \)-cobordism and (even-dimensional) surgery theorems. The topology of three-dimensional manifolds is closely tied to the fundamental group by the classical disk locating theorems, Dehn’s Lemma and the Loop Theorem. These theorems make the hierarchy theory run and eventually lead to toroidal decomposition. (And conversely, the least understood 3-manifolds are those having no fundamental group to decompose by imbedded disks — homotopy 3-spheres.) One could extend this pattern to dimension two by quoting the continuous-boundary-value Riemann mapping theorem (together with the uniformization theorem) as the 2-dimensional disk theorem.

There is now a 4-dimensional 2-disk imbedding theorem. Its simply connected version was the key to the work on the Poincaré conjecture [Freedman 1982]. The body of this paper is a discussion of its proof, with applications being given at the end.

Czesław Olech	Related
Zbigniew Ciesielski	Related

article M. H. Freedman: “There is no room to spare in four-dimensional space,” Notices Amer. Math. Soc. 31 : 1 (1984), pp. 3–6. MR 0728340 Zbl 0538.57001

incollection A. Casson and M. Freedman: “Atomic surgery problems,” pp. 181–199 in Four-manifold theory (Durham, NH, July 4–10, 1982). Edited by C. Gordon and R. C. Kirby. Contemporary Mathematics 35. American Mathematical Society (Providence, RI), 1984. MR 0780579 Zbl 0559.57008

Andrew J. Casson	Related
Robion C. Kirby	Related	Volume
Cameron McAllan Gordon	Related	Volume

article M. H. Freedman: “A new technique for the link slice problem,” Invent. Math. 80 : 3 (1985), pp. 453–465. MR 0791669 Zbl 0569.57002

article M. H. Freedman and L. R. Taylor: “A universal smoothing of four-space,” J. Differential Geom. 24 : 1 (1986), pp. 69–78. MR 0857376 Zbl 0586.57007

Except in dimension four, smooth structures can be classified up to \( \varepsilon \)-isotopy by bundle reductions. Since \( \mathbb{R}^n \) is contractible, this implies that any smooth structure \( \Gamma \) on \( \mathbb{R}^n \), \( n \neq 4 \), is \( \varepsilon \)-isotopic to the standard one. In contrast, \( \mathbb{R}^4 \) has many distinct smoothings (even up to diffeomorphism.) We construct a certain smoothing of the half-space, \[ \tfrac{1}{2}\mathbb{R}^4 = \{(x_1,x_2,x_3,x_4)|x_4\geq 0\} \] and write \( H \) for this half-space together with its smooth structure. \( H \) contains all other smoothings of \( \frac{1}{2}\mathbb{R}^4 \) and is unique with respect to this property. \( H \) is the universal half-space. The interior \( \mathring H = U \) is naturally identified (replace \( x_4 \) with \( \ln x_4 \)) with a smoothing of \( \mathbb{R}^4 \). Corollary B states that \( U \) contains every smoothing of \( \mathbb{R}^4 \) imbedded within it. Thus, we say \( U \) is a universal \( \mathbb{R}^4 \). The construction of \( U \) is unambiguous but we do not claim that any \( \mathbb{R}_{\Gamma}^4 \) into which all smooth \( \mathbb{R}^4 \)’s imbed is diffeomorphic to \( U \). This is not known.

incollection M. H. Freedman: “A geometric reformulation of 4-dimensional surgery,” pp. 133–141 in Special volume in honor of R. H. Bing (1914–1986), published as Topology Appl. 24 : 1–3 (1986). MR 0872483 Zbl 0898.57005

incollection M. H. Freedman: “Are the Borromean rings \( A \)-\( B \)-slice?,” pp. 143–145 in Special volume in honor of R. H. Bing (1914–1986), published as Topology Appl. 24 : 1–3. Elsevier Science B.V. (North-Holland) (Amsterdam), 1986. MR 0872484 Zbl 0627.57004

article “Michael H. Freedman awarded 1986 Veblen Prize,” Notices Amer. Math. Soc. 33 : 2 (1986), pp. 227–228. MR 830614

M. H. Freedman: Jon Loni’s Stoney Point massacre: A story twice retold, 1986.

misc M. F. Atiyah, B. Mazur, J. W. Milnor, and V. Strassen: Addresses on the work of the 1986 Fields Medalists and Nevanlinna Prize winner, 1987. 60-min. videocassette. American Mathematical Society (Providence, RI). Plenary addresses presented at the International Congress of Mathematicians (Berkeley, CA, August 1986). MR 1064525

Leslie G. Valiant	Related
M. F. Atiyah	Related	Volume
Gerd Faltings	Related
Volker Strassen	Related
Barry C. Mazur	Related
John W. Milnor	Related
S. K. Donaldson	Related

incollection J. Milnor: “The work of M.H. Freedman,” pp. 13–15 in Proceedings of the International Congress of Mathematicians 1986 (August 3–11, Berkeley, CA), vol. 2. Edited by A. Gleason. American Mathematical Society (Providence, RI), 1987. MR 0934211 Zbl 0659.01014

Andrew Mattei Gleason	Related	Volume
John W. Milnor	Related

article M. H. Freedman: “A power law for the distortion of planar sets,” Discrete Comput. Geom. 2 : 4 (1987), pp. 345–351. MR 0911188 Zbl 0642.52006

We consider how to map the sites of a square region of planar lattice into a three-dimensional cube, so as to minimize the maximum distortion of distance. We consider the cube to be endowed with a “foliated” geometry in which horizontal distance is standard but vertical communication only occurs at the surface of the cube. These geometries may naturally arise if a planar data set is to be stored in a stack of chips. It is proved that any one-to-one map which fills the cube with a fixed “density” must produce a distortion of distance which grows as the one-sixth power of the diameter of the square and the two-thirds power of the density. Moreover, we explicitly define one-to-one maps with \( 100\% \) density, one-sixth power stretching, and a small leading coefficient. As a final note, a high-dimensional analog is considered.

article M. H. Freedman and R. Skora: “Strange actions of groups on spheres,” J. Differential Geom. 25 : 1 (1987), pp. 75–98. MR 0873456 Zbl 0588.57024

article K. Kuga: “The contributions of Michael H. Freedman,” Sugaku 39 : 1 (1987), pp. 8–16. MR 904866

article M. H. Freedman: “A note on topology and magnetic energy in incompressible perfectly conducting fluids,” J. Fluid Mech. 194 (1988), pp. 549–551. MR 0988303 Zbl 0676.76095

article M. H. Freedman and Z.-X. He: “Factoring the logarithmic spiral,” Invent. Math. 92 : 1 (1988), pp. 129–138. MR 0931207 Zbl 0622.30011

article M. H. Freedman and Z.-X. He: “A remark on inherent differentiability,” Proc. Amer. Math. Soc. 104 : 4 (1988), pp. 1305–1310. MR 0937012 Zbl 0689.57021

article M. H. Freedman: “Whitehead\( {}_3 \) is a ‘slice’ link,” Invent. Math. 94 : 1 (1988), pp. 175–182. MR 0958596 Zbl 0678.57002

incollection M. H. Freedman and R. Skora: “Strange actions of groups on spheres, II,” pp. 41–57 in Holomorphic functions and moduli (MSRI, Berkeley, CA, March 13–19, 1986), vol. II. Edited by D. Drasin, C. J. Earle, F. W. Gehring, I. Kra, and A. Marden. Mathematical Sciences Research Institute Publications 11. Springer (New York), 1988. MR 0955832 Zbl 0666.57028

Irwin Kra	Related
Richard Kevin Skora	Related
David Drasin	Related
Clifford John Earle, Jr.	Related
Frederick W. Gehring	Related

article M. H. Freedman: “Poincaré transversality and four-dimensional surgery,” Topology 27 : 2 (1988), pp. 171–175. MR 0948180 Zbl 0654.57007

M. H. Freedman: A proposal on Panama, 1988.

article M. H. Freedman and X.-S. Lin: “On the \( (A,B) \)-slice problem,” Topology 28 : 1 (1989), pp. 91–110. MR 0991101 Zbl 0845.57016

book M. H. Freedman and F. Luo: Selected applications of geometry to low-dimensional topology (The Pennsylvania State University, University Park, PA, February 2–5, 1987), vol. 1. University Lecture Series 1. American Mathematical Society (Providence, RI), 1989. Marker Lectures in the Mathematical Sciences. MR 1043633 Zbl 0691.57001

book M. H. Freedman and F. Quinn: Topology of 4-manifolds. Princeton Mathematical Series 39. Princeton University Press (Princeton, NJ), 1990. MR 1201584 Zbl 0705.57001

M. H. Freedman: Night climbing, 1990.

article M. H. Freedman and Z.-X. He: “Links of tori and the energy of incompressible flows,” Topology 30 : 2 (1991), pp. 283–287. MR 1098922 Zbl 0731.57003

The moduli of curve families have been a useful bridge between analysis and geometric-topological argument. Caratheódory exploited the notion to construct his theory of prime ends. In dimensions three and higher this connection has been extensively developed by Fred Gehring (for example, see [Gehring 1971; 1986]). We continue in this tradition by showing that the naturally defined “conformal moduli” for a disjoint collection of solid tori in \( \mathbb{R}^3 \) cannot all be greater than the constant \( 125\pi/48 \) if the tori are linked in any essential manner. It is natural to conjecture that the optimal lower bound is \( (\sqrt{2}-1)/2\pi \), the modulus of the “solid” Clifford torus.

As an application, we use the topology of linking flow lines to estimate a lower bound on the energy of certain incompressible flows. Roughly, one thinks that an invariant solid torus of spinning fluid may give up energy by elongating like a soda straw, but that this should be prevented if several such tori are linked. To make this precise, an inequality relating modulus and a variant of energy is derived.

article M. H. Freedman and Z.-X. He: “Divergence-free fields: Energy and asymptotic crossing number,” Ann. of Math. (2) 134 : 1 (1991), pp. 189–229. MR 1114611 Zbl 0746.57011

article M. H. Freedman: “An unknotting result for complete minimal surfaces [in] \( \mathbf{R}^ 3 \),” Invent. Math. 109 : 1 (1992), pp. 41–46. MR 1168364 Zbl 0773.53003

incollection M. H. Freedman and Z.-X. He: “Research announcement on the ‘energy’ of knots,” pp. 219–222 in Topological aspects of the dynamics of fluids and plasmas (University of California at Santa Barbara, August–December 1991). Edited by H. K. Moffatt, G. M. Zaslavasky, P. Comte, and M. Tabor. NATO ASI Series E: Applied Sciences 218. Kluwer Academic Publishers (Dordrecht), 1992. MR 1232232 Zbl 0788.53004

Zheng-Xu He	Related
Pierre Comte	Related
Michael Tabor	Related
Henry Keith Moffatt	Related
George M. Zaslavsky	Related

article S. De Michelis and M. H. Freedman: “Uncountably many exotic \( \mathbf{R}^ 4 \)’s in standard 4-space,” J. Differential Geom. 35 : 1 (1992), pp. 219–254. MR 1152230 Zbl 0780.57012

It is known that the standard (Euclidean) smooth structure on 4-space when restricted to certain open subsets homeomorphic to \( \mathbb{R}^4 \) gives a smooth structure which is not diffeomorphic to the standard one. This behavior is a consequence of Donaldson’s counterexample [1987] to the smooth 5-dimensional \( h \)-cobordism theorem, and was noticed (in anticipation of Donaldson’s result) by A. Casson and the second named author (see [Kirby 1989]). Taubes [1987] developed a technically demanding theory of the Yang–Mills equation on “asymptotically end periodic” 4-manifolds in part to verify that a known family of exotic \( \mathbb{R}^4 \)’s were mutually distinct. That family lays smoothly in \( \mathbb{S}^2\times\mathbb{S}^2 \) but not \( \mathbb{R}^4 \). We combine ideas from the above-mentioned papers to address a nested family of \( \mathbb{R}^4 \) homeomorphs called “ribbon \( \mathbb{R}^4 \)’s” lying in \( \mathbb{R}^4 \) standard. There are continuum many pairwise distinct smooth structures represented within this family.

incollection M. H. Freedman: “Working and playing with the 2-disk,” pp. 37–47 in Mathematics into the twenty-first century: Proceedings of the 1988 Centennial Symposium (Providence, RI, August 8–12, 1988). Edited by F. E. Browder. American Mathematical Society Centennial Publications II. American Mathematical Society (Providence, RI), 1992. MR 1184613 Zbl 0924.57026

This article is simply a written lecture and what philosophy it contains should not necessarily be taken seriously. However, it is much easier to learn a whole story than a single theorem, so many of the latter are woven into the former. Our hero, for fun, is the two-dimensional disk which seems to intrude at many important junctures of geometric topology. Also, there is the theme that ideas of great importance can be enormously simple. As the Centennial recalls to each of us our small mortal places and seems to threaten even mathematics with a certain loss of youth — computer proofs, proofs too long to write (or think), the joint power and vacuity of abstraction — I enjoy recalling a few forceful but simple ideas in the subject I know best. I have no prediction for the next century but am content to express the hope that mathematics will still, from time to time, be extraordinarily easy — that the last simple idea is still far off.

incollection M. H. Freedman and M. A. Berger: “Combinatorial relaxation of magnetic fields,” pp. 91–96 in Magnetohydrodynamic stability and dynamos, published as Geophys. Astrophys. Fluid Dynam. 73 : 1–4 (1993). MR 1289021

article R. Stong: “Uniqueness of \( \pi_1 \)-negligible embeddings in 4-manifolds: A correction to Theorem 10.5 of Freedman and Quinn,” Topology 32 : 4 (1993), pp. 677–699. MR 1241868

Frank Stringfellow Quinn, III	Related
Richard Stong	Related

article S. Bryson, M. H. Freedman, Z.-X. He, and Z. Wang: “Möbius invariance of knot energy,” Bull. Amer. Math. Soc. (N.S.) 28 : 1 (1993), pp. 99–103. MR 1168514 Zbl 0776.57003 ArXiv math/9301212

Zhenghan Wang	Related
Zheng-Xu He	Related
Steve Bryson	Related

article M. H. Freedman: “Link compositions and the topological slice problem,” Topology 32 : 1 (1993), pp. 145–156. MR 1204412 Zbl 0782.57010

article M. H. Freedman and Z. Wang: “\( \mathbf{C}P^ 2 \)-stable theory,” Math. Res. Lett. 1 : 1 (1994), pp. 45–48. MR 1258488 Zbl 0849.57016

article M. H. Freedman, Z.-X. He, and Z. Wang: “Möbius energy of knots and unknots,” Ann. of Math. (2) 139 : 1 (1994), pp. 1–50. MR 1259363 Zbl 0817.57011

Zhenghan Wang	Related
Zheng-Xu He	Related

article M. H. Freedman, V. S. Krushkal, and P. Teichner: “Van Kampen’s embedding obstruction is incomplete for 2-complexes in \( \mathbf{R}^ 4 \),” Math. Res. Lett. 1 : 2 (March 1994), pp. 167–176. MR 1266755 Zbl 0847.57005

V. Slava Krushkal	Related
Peter Teichner	Related

incollection M. H. Freedman and Z. Wang: “Controlled linear algebra,” pp. 138–156 in Prospects in topology: Proceedings of a conference in honor of William Browder (Princeton, NJ, March 1994). Edited by F. Quinn. Annals of Mathematics Studies 138. Princeton University Press (Princeton, NJ), 1995. MR 1368657 Zbl 0924.19003

Zhenghan Wang	Related
Frank Stringfellow Quinn, III	Related
William Browder	Related

article M. H. Freedman and F. Luo: “Equivariant isotopy of unknots to round circles,” Topology Appl. 64 : 1 (1995), pp. 59–74. MR 1339758 Zbl 0830.53002

article M. H. Freedman and P. Teichner: “4-manifold topology I: Subexponential groups,” Invent. Math. 122 : 3 (1995), pp. 509–529. MR 1359602 Zbl 0857.57017

The technical lemma underlying the 5-dimensional topological \( s \)-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating \( \pi_1 \)-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (“good groups”) where the \( \pi_1 \)-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations: (1) extension and (2) direct limit. The finitely generated groups in thisclass are amenable and no amenable group is known to lie outside this class.

article M. H. Freedman and P. Teichner: “4-Manifold topology II: Dwyer’s filtration and surgery kernels,” Invent. Math. 122 : 1 (1995), pp. 531–557. MR 1359603 Zbl 0857.57018

article M. H. Freedman and W. H. Press: “Truncation of wavelet matrices: Edge effects and the reduction of topological control,” Linear Algebra Appl. 234 (1996), pp. 1–19. MR 1368768 Zbl 0843.65100

Edge effects and Gibbs phenomena are a ubiquitous problem in signal processing. We show how this problem can arise from a mismatch between the “topology” of the data \( D \) (e.g., an interval in the case of a time series or a rectangle in the case of a photographic image) and the topology \( X \) (often a circle or torus) natural to the construction of the transformation \( O \). The notion of a manifold control space \( X \) for an orthogonal transformation \( O \) is introduced. It is proved that no matter how complicated \( X \) is, \( O \) may be “truncated” to an \( O^{\prime} \) with control space \( D \), homeomorphic to an interval or a product of intervals. This yields a new, topologically motivated approach to edge effects. We give the complete details for applying this approach to the discrete Daubechies transform of functions on the unit interval so that no data are wrapped around from one end of the interval to the other.

article C. L. Curtis, M. H. Freedman, W. C. Hsiang, and R. Stong: “A decomposition theorem for \( h \)-cobordant smooth simply-connected compact 4-manifolds,” Invent. Math. 123 : 2 (1996), pp. 343–348. MR 1374205 Zbl 0843.57020

Let \( M \) and \( N \) be smooth, \( h \)-cobordant compact 1-connected 4-manifolds. There exist decompositions \[ M = M_0 \cup_{\Sigma}M_1 \quad\text{and}\quad N = N_0 \cup_{\Sigma}N_1 \] where \( M_0 \) and \( N_0 \) are smooth compact contractible 4-manifolds with boundary \( \Sigma \), so that \( (M_1,\Sigma) \), and \( (N_1,\Sigma) \) are diffeomorphic. If \( M \) and \( N \) are closed, then we may further arrange that \( M_1 \) and \( N_1 \) are 1-connected. In fact, if \( W \) is an \( h \)-cobordism connecting \( M \) and \( N \), then \( W \) can be written \[ W = W_0 \cup_{\Sigma\times I}W_1 \] where \( (W_0;M_0,N_0) \) is an (often) nontrivial \( h \)-cobordism and \( (W_1;M_1,N_1) \) is smoothly the product \( h \)-cobordism \[ (M_1\times I;\,M_1\times\{0\},\,M_1\times\{1\}) .\]

Wu-Chung Hsiang	Related
Cynthia Louise Curtis	Related
Richard Stong	Related

incollection J. Milnor: “The work of M. H. Freedman,” pp. 405–408 in Fields Medallists’ lectures. Edited by M. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622913

M. F. Atiyah	Related	Volume
Daniel Iagolnitzer	Related
John W. Milnor	Related

M. H. Freedman and V. S. Krushkal: “Notes on ends of hyperbolic 3-manifolds” in Thirteenth annual workshop in geometric topology (Colorado Springs, CO, June 13–15, 1996). The Colorado College, 1997. Informal publication of The Colorado College, 1997.

article M. H. Freedman: “Percolation on the projective plane,” Math. Res. Lett. 4 : 6 (1997), pp. 889–894. MR 1492127 Zbl 0902.60085

Since the projective plane is closed, the natural homological observable of a percolation process is the presence of the essential cycle in \( H_1(\mathbb{R}P^2;\mathbb{Z}_2) \). In the Voroni model at critical phase, \( p_c = .5 \), this observable has probability \( q = .5 \) independent of the metric on \( \mathbb{R}P^2 \). This establishes a single instance (\( \mathbb{R}P^2 \), homological observable) of a very general conjecture about the conformal invariance of percolation due to Aizenman and Langlands, for which there is much moral and numerical evidence but no previously verified instances. On \( \mathbb{R}P^2 \) all metrics are conformally equivalent so the proof of metric independence is precisely what the conjecture would predict. What is very special, is that at \( p_c \) metric invariance holds in all finite models so passing to the limit is trivial; the probability \( q \) is fixed at \( .5 \) by a topological symmetry.

incollection M. Freedman, R. Hain, and P. Teichner: “Betti number estimates for nilpotent groups,” pp. 413–434 in Fields Medallists’ lectures. Edited by M. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. MR 1622914

We prove an extension of the following result of Lubotzky and Magid on the rational cohomology of a nilpotent group \( G \): if \( b_1 < \infty \) and \( G\otimes\mathbb{Q} \neq 0 \), \( \mathbb{Q} \), \( \mathbb{Q}^2 \), then \( b_2 > b_1^2/4 \). Here the \( b_i \) are the rational Betti numbers of \( G \) and \( G\otimes\mathbb{Q} \) denotes the Malcev-completion of \( G \). In the extension, the bound is improved when we know that all relations of \( G \) all have at least a certain commutator length. As an application of the refined inequality, we show that each closed oriented 3-manifold falls into exactly one of the following classes: it is a rational homology 3-sphere, or it is a rational homology \( \mathbb{S}^1\times\mathbb{S}^2 \), or it has the rational homology of one of the oriented circle bundles over the torus (which are indexed by an Euler number \( n\in\mathbb{Z} \), e.g. \( n=0 \) corresponds to the 3-torus) or it is of general type by which we meant that the rational lower central series of the fundamental group does not stabilize. In particular, any 3-manifold group which allows a maximal torsion-free nilpotent quotient admits a rational homology isomorphism to a torsion-free nilpotent group.

M. F. Atiyah	Related	Volume
Peter Teichner	Related
Richard M. Hain	Related
Daniel Iagolnitzer	Related

article B. Freedman and M. H. Freedman: “Kneser–Haken finiteness for bounded 3-manifolds locally free groups, and cyclic covers,” Topology 37 : 1 (1998), pp. 133–147. MR 1480882 Zbl 0896.57012

Associated to every compact 3-manifold \( M \) and positive integer \( b \), there is a constant \( c(M,b) = c \). Any collection \( F_i \) of incompressible surfaces with Betti numbers \( b_1F_i < b \) for all \( i \), none of which is a boundary parallel annulus or a boundary parallel disk, and no two of which are parallel, must have fewer than \( c \) members. Our estimate for \( c \) is exponential in \( b \). This theorem is used to detect closed incompressible surfaces in the infinite cyclic covers of all non-fibered knot complements. In other terms, if the commutator subgroup of a knot group is locally free, then it is actually a finitely generated free group.

article M. H. Freedman: “Limit, logic, and computation,” Proc. Natl. Acad. Sci. USA 95 : 1 (1998), pp. 95–97. MR 1612421 Zbl 0891.68041

article M. H. Freedman: “P/NP, and the quantum field computer,” Proc. Natl. Acad. Sci. USA 95 : 1 (1998), pp. 98–101. MR 1612425 Zbl 0895.68053

The central problem in computer science is the conjecture that two complexity classes, \( P \) (polynomial) and \( NP \) (nondeterministic polynomial time — roughly those decision problems for which a proposed solution can be checked in polynomial time), are distinct in the standard Turing model of computation: \( P\neq NP \). As a generality, we propose that each physical theory supports computational models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of implementation of the Turing machine. Non-Abelian topological quantum field theories exhibit the mathematical features necessary to support a model capable of solving all \( \#P \) problems, a computationally intractable class, in polynomial time. Specifically [Witten 1989] has identified expectation values in a certain \( \mathit{SU}(2) \)-field theory with values of the Jones polynomial [Jones 1985] that are \( \#P \)-hard [Jaeger, Vertigen and Welsh 1990]. This suggests that some physical system whose effective Lagrangian contrains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving \( NP \) or even \( \#P \)-hard problems in polynomial time. Defining such a system and addressing the accuracy issues inherent in preparation and measurement is a major unsolved problem.

article M. H. Freedman and C. T. McMullen: “Elder siblings and the taming of hyperbolic 3-manifolds,” Ann. Acad. Sci. Fenn. Math. 23 : 2 (1998), pp. 415–428. MR 1642126 Zbl 0918.57004

incollection M. H. Freedman: “Topological views on computational complexity,” pp. 453–464 in Proceedings of the International Congress of Mathematicians (Berlin, 1998), published as Doc. Math. Extra II. Fakultät für Mathematik, Universität Bielefeld (Bielefeld), 1998. MR 1648095 Zbl 0967.68520

incollection M. H. Freedman: “\( K \)-sat on groups and undecidability,” pp. 572–576 in Proceedings of the thirtieth annual ACM symposium on theory of computing (Dallas, TX, May 23–26, 1998). Edited by Association for Computing Machinery. Association for Computing Machinery (New York), 1998. MR 1715605 Zbl 1028.68068

The general Boolean formula can be quickly converted into a normal form, \( N_3 \), the conjunction of triple disjunctions of literals, which is satisfiable iff the original is. On the other hand, formulae in \( N_2 \), the conjunction of 2-fold disjunctions of literals, can be checked for a satisfaction in polynomial time. Thus these two satisfaction problems, “2-sat” and “3-sat,” have been considered as an interesting boundary point between \( P \) and \( NP \). We define an infinite generalization of 2-sat and 3-sat which are respectively algorithmic and undecidable. As a corollary it is noted that the 3-colorability of doubly-periodic planar graphs is undecidable. It was suggested in [Freedman 1998] that a general approach to proving \( P\neq NP \) would be to construct some infinitary limit of decision problems with the property that those admitting polynomial time algorithms would be decidable in this limit. The hope here is to exploit the strong connection between polynomial growth and finite dimensionality. Since logic has a method — self-reference — for establishing problems as undecidable, this technique applied to the limit could potentially show that the finite-decision problem lies outside of \( P \). This paper supplies one way of extending \( k \)-sat to an infinite context in which decidability distinguishes 2-sat from 3-sat.

incollection M. H. Freedman: “Zeldovich’s neutron star and the prediction of magnetic froth,” pp. 165–172 in The Arnoldfest: Proceedings of a conference in honour of V. I. Arnold for his sixtieth birthday (Toronto, ON, June 15–21, 1997). Edited by E. Bierstone, B. Khesin, A. Khovanskii, and J. E. Marsden. Fields Institute Communications 24. American Mathematical Society (Providence, RI), 1999. MR 1733574 Zbl 0973.76097

Vladimir Igorevich Arnold	Related
Jerrold E. Marsden	Related
Edward Bierstone	Related
Boris A. Khesin	Related
Askold Georgievich Khovanskii	Related

incollection M. H. Freedman: “\( Z_2 \)-systolic-freedom,” pp. 113–123 in Proceedings of the Kirbyfest (MSRI, Berkeley, CA, June 22–26, 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. Mathematical Sciences Publishers, 1999. MR 1734404 Zbl 1024.53029 ArXiv math/0002124

Joel Hass	Related	Volume
Martin Scharlemann	Related	Volume
Robion C. Kirby	Related	Volume

article M. Freedman, H. Howards, and Y.-Q. Wu: “Extension of incompressible surfaces on the boundaries of 3-manifolds,” Pacific J. Math. 194 : 2 (2000), pp. 335–348. MR 1760785 Zbl 1015.57010 ArXiv math/9706222

Hugh Howards	Related
Ying-Qing Wu	Related

article M. H. Freedman and D. A. Meyer: “Projective plane and planar quantum codes,” Found. Comput. Math. 1 : 3 (2001), pp. 325–332. MR 1838758 Zbl 0995.94037 ArXiv quant-ph/9810055

article M. H. Freedman: “Quantum computation and the localization of modular functors,” Found. Comput. Math. 1 : 2 (2001), pp. 183–204. MR 1830035 Zbl 1004.57026 ArXiv quant-ph/0003128

article M. H. Freedman, M. J. Larsen, and Z. Wang: “The two-eigenvalue problem and density of Jones representation of braid groups,” Comm. Math. Phys. 228 : 1 (2002), pp. 177–199. MR 1911253 Zbl 1045.20027

Zhenghan Wang	Related
Michael J. Larsen	Related

article M. H. Freedman, M. Larsen, and Z. Wang: “A modular functor which is universal for quantum computation,” Comm. Math. Phys. 227 : 3 (2002), pp. 605–622. MR 1910833 Zbl 1012.81007 ArXiv quant-ph/0001108

Zhenghan Wang	Related
Michael J. Larsen	Related

article M. H. Freedman, A. Kitaev, and Z. Wang: “Simulation of topological field theories by quantum computers,” Comm. Math. Phys. 227 : 3 (2002), pp. 587–603. MR 1910832 Zbl 1014.81006 ArXiv quant-ph/0001071

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian \( H \) for a time \( t \). In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, \( H \equiv 0 \). These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:

TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.
TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

Zhenghan Wang	Related
Alexei Kitaev	Related

article M. H. Freedman: “Poly-locality in quantum computing,” Found. Comput. Math. 2 : 2 (2002), pp. 145–154. MR 1894373 Zbl 1075.81507 ArXiv quant-ph/0001077

A polynomial depth quantum circuit affects, by definition, a poly-local unitary transformation of a tensor product state space. It is a reasonable belief [Feynman 1982; Lloyd 1996; Freedman, Kitaev and Wang 2002] that, at a fine scale, these are precisely the transformations which will be available from physics to solve computational problems. The poly-locality of a discrete Fourier transform on cyclic groups is at the heart of Shor’s factoring algorithm. We describe a class of poly-local transformations, which include the discrete orthogonal wavelet transforms, in the hope that these may be helpful in constructing new quantum algorithms. We also observe that even a rather mild violation of poly-locality leads to a model without one-way functions, giving further evidence that poly-locality is an essential concept.

article M. H. Freedman, K. Walker, and Z. Wang: “Quantum \( \mathit{SU}(2) \) faithfully detects mapping class groups modulo center,” Geom. Topol. 6 (2002), pp. 523–539. MR 1943758 Zbl 1037.57024 ArXiv math.GT/0209150

The Jones–Witten theory gives rise to representations of the (extended) mapping class group of any closed surface \( Y \) indexed by a semi-simple Lie group \( G \) and a level \( k \). In the case \( G = \mathit{SU}(2) \) these representations (denoted \( V_A(Y) \)) have a particularly simple description in terms of the Kauffman skein modules with parameter \( A \) a primitive \( 4r \)-th root of unity (\( r = k + 2 \)). In each of these representations (as well as the general \( G \) case), Dehn twists act as transformations of finite order, so none represents the mapping class group \( \mathcal{M}(Y) \) faithfully. However, taken together, the quantum \( \mathit{SU}(2) \) representations are faithful on non-central elements of \( \mathcal{M}(Y) \). (Note that \( \mathcal{M}(Y) \) has non-trivial center only if \( Y \) is a sphere with 0, 1, or 2 punctures, a torus with 0, 1, or 2 punctures, or the closed surface of genus \( = 2 \).) Specifically, for a non-central \( h\in\mathcal{M}(Y) \) there is an \( r_0(h) \) such that if \( r \geq r_0(h) \) and \( A \) is a primitive \( 4r \)-th root of unity then \( h \) acts projectively nontrivially on \( V_A(Y) \). Jones’ [1987] original representation \( \rho_n \) of the braid groups \( B_n \), sometimes called the generic \( q \)-analog-\( \mathit{SU}(2) \)-representation, is not known to be faithful. However, we show that any braid \( h \neq \mathrm{id} \in B_n \) admits a cabling \( c = c_1,\dots,c_n \) so that \( \rho_N(c(h)) \neq\mathrm{id} \), \( N = c_1 + \dots +c_n \).

Kevin Walker	Related
Zhenghan Wang	Related

incollection M. H. Freedman, D. A. Meyer, and F. Luo: “\( Z_2 \)-systolic freedom and quantum codes,” pp. 287–320 in Mathematics of quantum computation. Edited by R. K. Brylinski and G. Chen. Computational Mathematics 3. Chapman & Hall/CRC (Boca Raton, FL), 2002. MR 2007952 Zbl 1075.81508

A closely coupled pair of conjectures/questions — one in differential geometry (by M. Gromov), the other in quantum information theory — are both answered in the negative. The answer derives from a certain metrical flexibility of manifolds and a corresponding improvement to the theoretical efficiency of existing local quantum codes. We exhibit this effect by constructing a family of metrics on \( \mathbb{S}^2\times\mathbb{S}^1 \), and other three and four dimensional manifolds. Quantitatively, the explicit “freedom” exhibited is too weak (a \( \log^{1/2} \) factor in the natural scaling) to yield practical codes but we cannot rule out the possibility of other families of geometries with more dramatic freedom.

Goong Chen	Related
Feng Luo	Related
David A. Meyer	Related
Ranee K. Brylinski	Related

article M. H. Freedman: “A magnetic model with a possible Chern–Simons phase,” Comm. Math. Phys. 234 : 1 (2003), pp. 129–183. With an appendix by F. Goodman and H. Wenzl. MR 1961959 Zbl 1060.81054 ArXiv quant-ph/0110060

An elementary family of local Hamiltonians \( H_{\circ,\ell} \), \( \ell = 1, 2, 3,\dots\, \), is described for a 2-dimensional quantum mechanical system of spin \( = 1/2 \) particles. On the torus, the ground state space \( G_{\circ,\ell} \) is (log) extensively degenerate but should collapse under “perturbation” to an anyonic system with a complete mathematical description: the quantum double of the \( \mathit{SO}(3) \)-Chern–Simons modular functor at \( q = e^{2\pi i/\ell+2} \) which we call \( \mathit{DE}\ell \). The Hamiltonian \( H_{\circ,\ell} \) defines a quantum loop gas. We argue that for \( \ell = 1 \) and 2, \( G_{\circ,\ell} \) is unstable and the collapse to \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \) can occur truly by perturbation. For \( \ell\geq 3 \), \( G_{\circ,\ell} \) is stable and in this case finding \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \) must require either \( \varepsilon > \varepsilon_\ell > 0 \), help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes “ ”. A hypothetical phase diagram is included in the introduction.

The effect of perturbation is studied algebraically: the ground state space \( G_{\circ,\ell} \) of \( H_{\circ,\ell} \) is described as a surface algebra and our ansatz is that perturbation should respect this structure yielding a perturbed ground state \( G_{\varepsilon,\ell} \) described by a quotient algebra. By classification, this implies \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \). The fundamental point is that nonlinear structures may be present on degenerate eigenspaces of an initial \( H_{\circ} \) which constrain the possible effective action of a perturbation.

There is no reason to expect that a physical implementation of \( G_{\varepsilon,\ell} \cong \mathit{DE}\ell \) as an anyonic system would require the low temperatures and time asymmetry intrinsic to Fractional Quantum Hall Effect (FQHE) systems or rotating Bose–Einstein condensates — the currently known physical systems modeled by topological modular functors. A solid state realization of \( \mathit{DE}3 \), perhaps even one at a room temperature, might be found by building and studying systems, “quantum loop gases,” whose main term is \( H_{\circ,3} \). This is a challenge for solid state physicists of the present decade. For \( l\geq 3 \), \( \ell\neq 2\mod 4 \), a physical implementation of \( \mathit{DE}\ell \) would yield an inherently fault-tolerant universal quantum computer. But a warning must be posted, the theory at \( \ell = 2 \) is not computationally universal and the first universal theory at \( \ell = 3 \) seems somewhat harder to locate because of the stability of the corresponding loop gas. Does nature abhor a quantum computer?

Frederick Michael Goodman	Related
Hans G. Wenzl	Related

article M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang: “Topological quantum computation,” Bull. Amer. Math. Soc. (N.S.) 40 : 1 (2003), pp. 31–38. MR 1943131 Zbl 1019.81008 ArXiv quant-ph/0101025

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten–Chern–Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and \( 2D \)-magnets are modeled by modular functors, opening a new possibility for the realization of quantum computers. The chief advantage of anyonic computation would be physical error correction: An error rate scaling like \( e^{-\alpha l} \), where \( l \) is a length scale, and \( \alpha \) is some positive constant. In contrast, the “presumptive” qubit-model of quantum computation, which repairs errors combinatorically, requires a fantastically low initial error rate (about \( 10^{-4} \)) before computation can be stabilized.

Zhenghan Wang	Related
Michael J. Larsen	Related
Alexei Kitaev	Related

techreport M. H. Freedman, C. Nayak, and K. Shtengel: Non-Abelian topological phases in an extended Hubbard model. Preprint, September 2003. ArXiv cond-mat/0309120

We describe four closely related Hubbard-like models (models A, B, C and D) of particles that can hop on a \( 2D \) Kagome lattice interacting via Coulomb repulsion. The particles can be either bosons (models A and B) or (spinless) fermions (models C and D). Models A and C also include a ring exchange term. In all four cases we solve equations in the model parameters to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate “\( d \)-isotopy space” \( \bar{V}_d \), \( 0 < d < 2 \). Near the “special” values, \( d = 2 \cos \pi/k+2 \), \( \bar{V}_d \) should collapse to a stable topological phase with anyonic excitations closely related to \( \mathit{SU}(2) \) Chern–Simons theory at level \( k \). We mention simplified models \( A^- \) and \( C^- \) which may also lead to these topological phases.

Kirill Shtengel	Related
Chetan Nayak	Related

article M. H. Freedman, A. Kitaev, and J. Lurie: “Diameters of homogeneous spaces,” Math. Res. Lett. 10 : 1 (2003), pp. 11–20. MR 1960119 Zbl 1029.22031 ArXiv quant-ph/0209113

Jacob Lurie	Related
Alexei Kitaev	Related

article M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang: “A class of \( P,T \)-invariant topological phases of interacting electrons,” Ann. Physics 310 : 2 (2004), pp. 428–492. MR 2044743 Zbl 1057.81053

We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in \( 2+1 \)-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. \( P \) and \( T \) invariance are maintained by a ‘doubling’ of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-\( T_c \) superconductivity. One is the ‘doubled’ version of the chiral spin liquid. The chiral spin liquid gives rise to anyon superconductivity at finite doping and the corresponding field theory is \( U(1) \) Chern–Simons theory at coupling constant \( m=2 \). The ‘doubled’ theory is two copies of this theory, one with \( m=2 \) the other with \( m=-2 \). The second example corresponds to \( Z_2 \) gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly organized discrete family indexed by the Baraha numbers, \( 2\cos(\pi/(k+2)) \), for positive integer \( k \). The physical inference is that a material manifesting the \( Z_2 \) gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.

Kirill Shtengel	Related
Zhenghan Wang	Related
Kevin Walker	Related
Chetan Nayak	Related

article M. H. Freedman, A. Kitaev, C. Nayak, J. K. Slingerland, K. Walker, and Z. Wang: “Universal manifold pairings and positivity,” Geom. Topol. 9 (2005), pp. 2303–2317. MR 2209373 Zbl 1129.57035 ArXiv math/0503054

Gluing two manifolds \( M_1 \) and \( M_2 \) with a common boundary \( S \) yields a closed manifold \( M \). Extending to formal linear combinations \( x = \sum a_iM_i \) yields a sesquilinear pairing \( p = \langle\,\cdot\,,\cdot\,\rangle \) with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing \( p \) onto a finite dimensional quotient pairing \( q \) with values in \( \mathbb{C} \) which in physically motivated cases is positive definite. To see if such a “unitary” TQFT can potentially detect any nontrivial \( x \), we ask if \( \langle x,x\rangle\neq 0 \) whenever \( x\neq 0 \). If this is the case, we call the pairing \( p \) positive. The question arises for each dimension \( d = 0,1,2,\dots\, \) We find \( p(d) \) positive for \( d = 0,1 \), and 2 and not positive for \( d = 4 \). We conjecture that \( p(3) \) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly \( s \)-cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for \( d = 3 + 1 \). There is a further physical implication of this paper. Whereas 3-dimensional Chern–Simons theory appears to be well-encoded within 2-dimensional quantum physics, e.g. in the fractional quantum Hall effect, Donaldson–Seiberg–Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.

Zhenghan Wang	Related
Johannes K. Slingerland	Related
Kevin Walker	Related
Chetan Nayak	Related
Alexei Kitaev	Related

article M. Bordewich, M. Freedman, L. Lovász, and D. Welsh: “Approximate counting and quantum computation,” Combin. Probab. Comput. 14 : 5–6 (2005), pp. 737–754. MR 2174653 Zbl 1089.68040

Motivated by the result that an ‘approximate’ evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes \( \# \)P and GapP have such an approximation scheme under certain natural normalizations. However, we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

D. Welsh	Related
Magnus Bordewich	Related
László Lovász	Related

M. H. Freedman, C. Nayak, and K. Shtengel: “Line of critical points in \( 2+1 \) dimensions: Quantum critical loop gases and non-abelian gauge theory,” Phys. Rev. Lett. 94 : 14 (2005), pp. 147205.

In this Letter, we
(1) construct a one-parameter family of lattice models of interacting spins;
(2) obtain their exact ground states;
(3) derive a statistical-mechanical analogy which relates their ground states to \( O(n) \) loop gases;
(4) show that the models are critical for \( d\leq\sqrt{2} \), where \( d \) parametrizes the models;
(5) note that, for the special values \( d=2\cos[\pi/(k+2)] \), they are related to doubled level-\( k \) \( \mathit{SU}(2) \) Chern–Simons theory;
(6) conjecture that they are in the universality class of a nonrelativistic \( \mathit{SU}(2) \) gauge theory; and
(7) show that its one-loop \( \beta \) function vanishes for all values of the coupling constant, implying that it is also on a critical line.

Kirill Shtengel	Related
Chetan Nayak	Related

article D. Das Sarma, M. H. Freedman, and C. Nayak: “Topologically-protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94 : 6 (2005), pp. 166802. ArXiv cond-mat/0412343

The Pfaffian state is an attractive candidate for the observed quantized Hall plateau at a Landau-level filling fraction \( \nu = 5/2 \). This is particularly intriguing because this state has unusual topological properties, including quasiparticle excitations with non-Abelian braiding statistics. In order to determine the nature of the \( \nu = 5/2 \) state, one must measure the quasiparticle braiding statistics. Here, we propose an experiment which can simultaneously determine the braiding statistics of quasiparticle excitations and, if they prove to be non-Abelian, produce a topologically protected qubit on which a logical Not operation is performed by quasiparticle braiding. Using the measured excitation gap at \( \nu = 5/2 \), we estimate the error rate to be \( 10^{-30} \) or lower.

Sankar Das Sarma	Related
Chetan Nayak	Related

M. Freedman, C. Nayak, and K. Shtengel: “An extended Hubbard model with ring exchange: A route to a non-abelian topological phase,” Phys. Rev. Lett. 94 : 6 (2005), pp. 066401.

Kirill Shtengel	Related
Chetan Nayak	Related

techreport M. Freedman, C. Nayak, and K. Walker: Tilted interferometry realizes universal quantum computation in the Ising TQFT without overpasses. Preprint, December 2005. ArXiv cond-mat/0512072

Kevin Walker	Related
Chetan Nayak	Related

article M. Freedman and V. Krushkal: “On the asymptotics of quantum \( \mathit{SU}(2) \) representations of mapping class groups,” Forum Math. 18 : 2 (2006), pp. 293–304. MR 2218422 Zbl 1120.57014

We investigate the rigidity and asymptotic properties of quantum \( \mathit{SU}(2) \) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum \( \mathit{SU}(2) \) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan’s property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of \( SL(2,\mathbb{Z}) \) on \( L^2(\mathbb{R}) \). As a consequence we obtain a curious analytic fact about the Fourier transform on the real line which may not have been previously observed.

M. Freedman, C. Nayak, and K. Walker: “Towards universal topological quantum computation in the \( \nu=5/2 \) fractional quantum Hall state,” Phys. Rev. B 73 : 24 (2006), pp. 245307.

The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction \( \nu = 5/2 \), can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, albeit with some operations which require error correction. On the other hand, if certain topology-changing operations can be implemented, then fully topologically-protected universal quantum computation is possible. In order to accomplish this, it is necessary to measure the interference between quasiparticle trajectories which encircle other moving trajectories in a time-dependent Hall droplet geometry [cond-mat/0512072].

Kevin Walker	Related
Chetan Nayak	Related

M. Freedman, S. Das Sarma, and C. Nayak: “Topological quantum computation,” Physics Today 59 : 7 (July 2006), pp. 32–38.

Sankar Das Sarma	Related
Chetan Nayak	Related

article D. Calegari and M. H. Freedman: “Distortion in transformation groups,” Geom. Topol. 10 (2006), pp. 267–293. With an appendix by Yves de Cornulier. MR 2207794 Zbl 1106.37017 ArXiv math/0509701

Yves de Cornulier	Related
Danny M. C. Calegari	Related

article S. H. Simon, N. E. Bonesteel, M. H. Freedman, N. Petrovic, and L. Hormozi: “Topological quantum computing with only one mobile quasiparticle,” Phys. Rev. Lett. 96 : 7 (2006), pp. 070503. MR 2205654 ArXiv quant-ph/0509175

Steven H. Simon	Related
Nicholas E. Bonesteel	Related
Layla Hormozi	Related
Nikola Petrovic	Related

article M. Freedman, L. Lovász, and A. Schrijver: “Reflection positivity, rank connectivity, and homomorphism of graphs,” J. Amer. Math. Soc. 20 : 1 (2007), pp. 37–51. MR 2257396 Zbl 1107.05089 ArXiv math/0404468

Alexander Schrijver	Related
László Lovász	Related

article M. Freedman, A. Feiguin, S. Trebst, A. Ludwig, M. Troyer, A. Kitaev, and Z. Wang: “Interacting anyons in topological quantum liquids: The golden chain,” Phys. Rev. Lett. 98 (2007), pp. 160409. ArXiv cond-mat/0612341

We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (“identity”) channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent \( z=1 \), and described by a two-dimensional (\( 2D \)) conformal field theory with central charge \( c=7/10 \). An exact mapping of the anyonic chain onto the \( 2D \) tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley–Lieb algebra. The gaplessness of the chain is shown to have topological origin.

Zhenghan Wang	Related
Adrian Esteban Feiguin	Related
A. Ludwig	Related
Simon Trebst	Related
Matthias Troyer	Related
Alexei Kitaev	Related

article M. H. Freedman and Z. Wang: “Large quantum Fourier transforms are never exactly realized by braiding conformal blocks,” Phys. Rev. A (3) 75 : 3 (2007), pp. 032322. MR 2312110 ArXiv cond-mat/0609411

Fourier transform is an essential ingredient in Shor’s factoring algorithm. In the standard quantum circuit model with the gate set \( \{U(2) \), controlled-NOT\( \} \), the discrete Fourier transforms \( F_N=(\omega^{ij})_{N\times N} \) for \( i,j=0,1,\dots,N{-}1 \) and \( \omega=e^{2\pi i/N} \) can be realized exactly by quantum circuits of size \( O(n^2) \) with \( n=\ln N \), and so can the discrete sine or cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci \( (2+1) \)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms \( F_N \) and the discrete sine or cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that an approximation is unavoidable in the implementation of Fourier transforms by braiding conformal blocks.

article M. H. Freedman and D. Gabai: “Covering a nontaming knot by the unlink,” Algebr. Geom. Topol. 7 (2007), pp. 1561–1578. MR 2366171 Zbl 1158.57024

article P. Bonderson, M. Freedman, and C. Nayak: “Measurement-only topological quantum computation,” Phys. Rev. Lett. 101 : 1 (2008), pp. 010501. MR 2429542 Zbl 1228.81121 ArXiv 0802.0279

Parsa Bonderson	Related
Chetan Nayak	Related

article C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma: “Non-abelian anyons and topological quantum computation,” Rev. Modern Phys. 80 : 3 (2008), pp. 1083–1159. MR 2443722 Zbl 1205.81062 ArXiv 0707.1889

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the \( \nu =5/2 \) state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the \( \nu =5/2 \) fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Steven H. Simon	Related
Ady Stern	Related
Chetan Nayak	Related
Sankar Das Sarma	Related

incollection M. Freedman: “Freedman’s writings on Lin,” pp. 430–431 in Topology and physics (Tianjin, China, 27–31 July 2007). Edited by K. Lin, Z. Weng, and W. Zhang. Nankai Tracts in Mathematics 12. World Scientific (Hackensack, NJ), 2008. Several letters written by Freedman to Lin. MR 2489601

Zhenghan Wang	Related
Kevin Lin	Related
Weiping Zhang	Related
Xiao-Song Lin	Related

M. Freedman, C. Nayak, and K. Shtengel: “Lieb–Schultz–Mattis theorem for quasitopological systems,” Phys. Rev. B 78 (2008), pp. 174411.

In this paper we address the question of the existence of a spectral gap in a class of local Hamiltonians. These Hamiltonians have the following properties: their ground states are known exactly; all equal-time correlation functions of local operators are short-ranged; and correlation functions of certain nonlocal operators are critical. A variational argument shows gaplessness with \( \omega \propto k^2 \) at critical points defined by the absence of certain terms in the Hamiltonian, which is remarkable because equal-time correlation functions of local operators remain short ranged. We call such critical points, in which spatial and temporal scaling are radically different, quasitopological. When these terms are present in the Hamiltonian, the models are in gapped topological phases which are of special interest in the context of topological quantum computation.

Kirill Shtengel	Related
Chetan Nayak	Related

incollection M. Freedman, C. Nayak, K. Walker, and Z. Wang: “On picture \( (2+1) \)-TQFTs,” pp. 19–106 in Topology and physics (Tianjin, China, 27–31 July 2007). Edited by K. Lin, Z. Weng, and W. Zhang. Nankai Tracts in Mathematics 12. World Scientific (Hackensack, NJ), 2008. MR 2503392 Zbl 1168.81024 ArXiv 0806.1926

Zhenghan Wang	Related
Kevin Lin	Related
Kevin Walker	Related
Chetan Nayak	Related
Weiping Zhang	Related
Xiao-Song Lin	Related

techreport M. H. Freedman: A topological phase in a quantum gravity model. Preprint, December 2008. A talk at Solvay conference, October 2008. ArXiv 0812.2278

I would claim that we do not have a suitably general definition of what a topological phase is, or more importantly, any robust understanding of how to enter one even in the world of mathematical models. The latter is, of course, the more important issue and the main subject of this note. But a good definition can sharpen our thinking and a poor definition can misdirect us. I will not attempt a final answer here but merely comment on the strengths and weaknesses of possible definitions and argue for some flexibility. In particular, I describe a rather simple class of “quantum gravity” models which are neither lattice nor field theoretic but appear to contain strong candidates for topological phases.

article P. Bonderson, M. Freedman, and C. Nayak: “Measurement-only topological quantum computation via anyonic interferometry,” Ann. Physics 324 : 4 (2009), pp. 787–826. MR 2508474 Zbl 1171.81004 ArXiv 0808.1933

We describe measurement-only topological quantum computation using both projective and interferometrical measurement of topological charge. We demonstrate how anyonic teleportation can be achieved using “forced measurement” protocols for both types of measurement. Using this, it is shown how topological charge measurements can be used to generate the braiding transformations used in topological quantum computation, and hence that the physical transportation of computational anyons is unnecessary. We give a detailed discussion of the anyonics for implementation of topological quantum computation (particularly, using the measurement-only approach) in fractional quantum Hall systems.

Parsa Bonderson	Related
Chetan Nayak	Related

article M. H. Freedman: “Complexity classes as mathematical axioms,” Ann. of Math. (2) 170 : 2 (2009), pp. 995–1002. MR 2552117 Zbl 1178.03069 ArXiv 0810.0033

article L. Fidkowski, M. Freedman, C. Nayak, K. Walker, and Z. Wang: “From string nets to nonabelions,” Comm. Math. Phys. 287 : 3 (2009), pp. 805–827. MR 2486662 Zbl 1196.82072 ArXiv cond-mat/0610583

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an \( \mathit{SO}(3)_3\times\mathit{SO}(3)_3 \) doubled Chern–Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.

Zhenghan Wang	Related
Lukasz Fidkowski	Related
Kevin Walker	Related
Chetan Nayak	Related

techreport P. Bonderson, S. Das Sarma, M. Freedman, and C. Nayak: A blueprint for a topologically fault-tolerant quantum computer. Preprint, March 2010. ArXiv 1003.2856

The advancement of information processing into the realm of quantum mechanics promises a transcendence in computational power that will enable problems to be solved which are completely beyond the known abilities of any “classical” computer, including any potential non-quantum technologies the future may bring. However, the fragility of quantum states poses a challenging obstacle for realization of a fault-tolerant quantum computer. The topological approach to quantum computation proposes to surmount this obstacle by using special physical systems — non-Abelian topologically ordered phases of matter — that would provide intrinsic fault-tolerance at the hardware level. The so-called “Ising-type” non-Abelian topological order is likely to be physically realized in a number of systems, but it can only provide a universal gate set (a requisite for quantum computation) if one has the ability to perform certain dynamical topology-changing operations on the system. Until now, practical methods of implementing these operations were unknown. Here we show how the necessary operations can be physically implemented for Ising-type systems realized in the recently proposed superconductor-semiconductor and superconductor-topological insulator heterostructures. Furthermore, we specify routines employing these methods to generate a computationally universal gate set. We are consequently able to provide a schematic blueprint for a fully topologically-protected Ising based quantum computer using currently available materials and techniques. This may serve as a starting point for attempts to construct a fault-tolerant quantum computer, which will have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and — most importantly — those applications which no one in our classical era has the prescience to foresee.

Parsa Bonderson	Related
Sankar Das Sarma	Related
Chetan Nayak	Related

techreport M. Freedman: Quantum gravity via manifold positivity. Preprint, August 2010. ArXiv 1008.1045

The macroscopic dimensions of space should not be input but rather output of a general model for physics. Here, dimensionality arises from a recently discovered mathematical bifurcation: positive versus indefinite manifold pairings. It is used to build an action on a formal chain of combinatorial space-times of arbitrary dimension. The context for such actions is 2-field theory where Feynman integrals are not over classical, but previously quantized configurations. A topologically enforced singularity of the action terminates the dimension at four and, in fact, the final fourth dimension is Lorentzian due to light-like vectors in the four dimensional manifold pairing. Our starting point is the action of causal dynamical triangulations but in a dimension-agnostic setting. It is encouraging that some hint of extra small dimensions emerges from our action.

article M. Freedman and V. Krushkal: “Topological arbiters,” J. Topol. 5 : 1 (February 2010), pp. 226–247. ArXiv 1002.1063

This paper initiates the study of topological arbiters, a concept rooted in Poincaré–Lefschetz duality. Given an \( n \)-dimensional manifold \( W \), a topological arbiter associates a value 0 or 1 to codimension-0 submanifolds of \( W \), subject to natural topological and duality axioms. For example, there is a unique arbiter on \( \mathbb{R}P^2 \), which reports the location of the essential 1-cycle. In contrast, we show that there exists an uncountable collection of topological arbiters in dimension 4. Families of arbiters, not induced by homology, are also shown to exist in higher dimensions. The technical ingredients underlying the 4-dimensional results are secondary obstructions to generalized link-slicing problems. For classical links in \( \mathbb{S}^3 \) the construction relies on the existence of nilpotent embedding obstructions in dimension 4, reflected in particular by the Milnor group. In higher dimensions novel arbiters are produced using non-trivial squares in stable homotopy theory.

The concept of “topological arbiter” derives from percolation and from 4-dimensional surgery. It is not the purpose of this paper to advance either of these subjects, but rather to study the concept for its own sake. However, in the appendices we give both an application to percolation, and the current understanding of the relationship between arbiters and surgery. An appendix also introduces a more general notion of a multi-arbiter. Properties and applications are discussed, including a construction of non-homological multi-arbiters.

article D. Calegari, M. H. Freedman, and K. Walker: “Positivity of the universal pairing in 3 dimensions,” J. Amer. Math. Soc. 23 : 1 (2010), pp. 107–188. MR 2552250 Zbl 1201.57024 ArXiv 0802.3208

Associated to a closed, oriented surface \( S \) is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along \( S \) defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary \( (2+1) \)-dimensional TQFTs.

The proof involves the construction of a suitable complexity function \( c \) on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy–Schwarz inequality, namely that \[ c(AB)\le \max(c(AA),c(BB)) \] for all \( A,B \) which bound \( S \), with equality if and only if \( A=B \).

The complexity function \( c \) involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol–Storm–Thurston.

Kevin Walker	Related
Danny M. C. Calegari	Related

article M. Freedman, R. Gompf, S. Morrison, and K. Walker: “Man and machine thinking about the smooth 4-dimensional Poincaré conjecture,” Quantum Topol. 1 : 2 (2010), pp. 171–208. MR 2657647 Zbl 1236.57043

While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincaré conjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen’s \( s \)-invariant, a slice obstruction within the general framework of Khovanov homology, changes this state of affairs. We studied a class of knots \( K \) for which nonzero \( s(K) \) would yield a counterexample to SPC4. Computations are extremely costly and we had only completed two tests for those \( K \), with the computations showing that \( s \) was 0, when a landmark posting of Akbulut [2009] altered the terrain. His posting, appearing only six days after our initial posting, proved that the family of “Cappell–Shaneson” homotopy spheres that we had geared up to study were in fact all standard. The method we describe remains viable but will have to be applied to other examples. Akbulut’s work makes SPC4 seem more plausible, and in another section of this paper we explain that SPC4 is equivalent to an appropriate generalization of Property R (“in \( \mathbb{S}^3 \), only an unknot can yield \( \mathbb{S}^1\times\mathbb{S}^2 \) under surgery”). We hope that this observation, and the rich relations between Property R and ideas such as taut foliations, contact geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to look at SPC4.

Robert E. Gompf	Related
Kevin Walker	Related
Scott Morrison	Related

article M. H. Freedman: “Group width,” Math. Res. Lett. 18 : 3 (2011), pp. 433–436. MR 2802577 ArXiv 1011.2460

article M. H. Freedman, L. Gamper, C. Gils, S. V. Isakov, S. Trebst, and M. Troyer: “Topological phases: An expedition off lattice,” Ann. Physics 326 : 8 (2011), pp. 2108–2137. MR 2812881 Zbl 1221.81219 ArXiv 1102.0270

Motivated by the goal to give the simplest possible microscopic foundation for a broad class of topological phases, we study quantum mechanical lattice models where the topology of the lattice is one of the dynamical variables. However, a fluctuating geometry can remove the separation between the system size and the range of local interactions, which is important for topological protection and ultimately the stability of a topological phase. In particular, it can open the door to a pathology, which has been studied in the context of quantum gravity and goes by the name of ‘baby universe’, Here we discuss three distinct approaches to suppressing these pathological fluctuations. We complement this discussion by applying Cheeger’s theory relating the geometry of manifolds to their vibrational modes to study the spectra of Hamiltonians. In particular, we present a detailed study of the statistical properties of loop gas and string net models on fluctuating lattices, both analytically and numerically.

Lukas Gamper	Related
Simon Trebst	Related
Matthias Troyer	Related
Charlotte Gils	Related
Sergei V. Isakov	Related

techreport S. J. Yamamoto, M. Freedman, and K. Yang: 3D non-abelian anyons: Degeneracy splitting and detection by adiabatic cooling. Preprint, February 2011. ArXiv 1102.5742

3D non-abelian anyons have been theoretically proposed to exist in heterostructures composed of type II superconductors and topological insulators. We use realistic material parameters for a device derived from \( \mathrm{Bi}_2\mathrm{Se}_3 \) to quantitatively predict the temperature and magnetic field regimes where an experiment might detect the presence of these exotic states by means of a cooling effect. Within the appropriate parameter regime, an adiabatic increase of the magnetic field will result in a decrease of system temperature when anyons are present. If anyons are not present, the same experiment would result in heating.

Kun Yang	Related
S. J. Yamamoto	Related

article M. Freedman, M. B. Hastings, C. Nayak, X.-L. Qi, K. Walker, and Z. Wang: “Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions,” Phys. Rev. B 83 : 11 (2011), pp. 115132. ArXiv 1005.0583

In a recent paper, Teo and Kane Phys. Rev. Lett. 104 046401 (2010) proposed a three-dimensional (3D) model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set \( R \) of unitary transformations on the zero-mode Hilbert space which is a “ghostly” recollection of the action of the braid group on Ising anyons in two dimensions. In this paper, we find the group \( T_{2n} \), which governs the statistics of these defects by analyzing the topology of the space \( K_{2n} \) of configurations of \( 2n \) defects in a slowly spatially varying gapped free-fermion Hamiltonian: \( T_{2n}\equiv \pi_1(K_{2n}) \). We find that the group \( T_{2n}=\mathbb{Z}\times T_{2n}^r \), where the “ribbon permutation group” \( T_{2n}^r \) is a mild enhancement of the permutation group \[ S_{2n}: T_{2n}^r\equiv \mathbb{Z}_2\rtimes E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) .\] Here, \( E((\mathbb{Z}_2)^{2n}\rtimes S_{2n}) \) is the “even part” of \( (\mathbb{Z}_2)^{2n}\rtimes S_{2n} \), namely, those elements for which the total parity of the element in \( (\mathbb{Z}_2)^{2n} \) added to the parity of the permutation is even. Surprisingly, \( R \) is only a projective representation of \( T_{2n} \), a possibility proposed by Wilczek [hep-th/9806228]. Thus, Teo and Kane’s defects realize projective ribbon permutation statistics, which we show to be consistent with locality. We extend this phenomenon to other dimensions, codimensions, and symmetry classes. We note that our analysis applies to 3D networks of quantum wires supporting Majorana fermions; thus, these networks are not required to be planar. Because it is an essential input for our calculation, we review the topological classification of gapped free-fermion systems and its relation to Bott periodicity.

Zhenghan Wang	Related
Kevin Walker	Related
Matthew B. Hastings	Related
Chetan Nayak	Related
Xiao-Liang Qi	Related

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