Celebratio Mathematica — Freedman

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

[2] 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

[3] 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.

[4] 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.

[5] 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

[6] 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.

[7] 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

[8] 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

[9]M. H. Freedman: [Unpublished handwritten notes].

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

[11] 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.

[12] 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

[13] 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.

[14]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

[15]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

[16] 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

[17] 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

[18] 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

[19] 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.

[20] 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

[21] 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

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

[23] 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

[24] 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

[25] 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.

[26] 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.

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

[28] 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

[29] 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

[30] 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.

[31] 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

[32] 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

[33] 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

[34] 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.

[35] 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

Topology in dImension 4