Celebratio Mathematica

[1] 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

[2] 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

[3] 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

[4] 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

[5]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.

[6] 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.

[7] 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

[8] 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

[9] 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

[10] 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

[11] 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.

[12] 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

[13] 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

[14] 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.

[15] 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

Michael H. Freedman

Topology in dimension 3