Celebratio Mathematica — Freedman

Topology in dImension 4

Works connected to Kevin Walker

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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. 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