Celebratio Mathematica

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

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.

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 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

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Michael H. Freedman

Michael Hartley Freedman

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