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Michael Freedman

Publications and source records attributed to Michael Freedman.

At least 19 recordsLinked to original sources
arXiv2026

Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold

Let \(M\) be a closed connected oriented topological \(4\)-manifold. We prove that if \(F_1,\dots,F_r\subset M\) are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera \(g_i\), same-sign twisted normal Euler numbers \(e_i\), and \( [F_1]+\cdots+[F_r]=0\in H_2(M;\F_2), \) then the normal-Euler excess \( \sum_{i=1}^r \bigl(\abs{e_i}-2g_i\bigr) \) is bounded above by a constant depending only on \(M\). Thus same-sign mod-\(2\)-null families of disjoint nonorientable surfaces in a fixed ambient \(4\)-manifold have uniformly bounded total excess over Massey's \(S^4\) bound. The proof combines a tubing construction with the signature and Euler-characteristic formulas for \(2\)-fold branched covers. As corollaries, every closed oriented topological \(4\)-manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of \(\RP^2\) with \(\abs{e}>2\), and only finitely many pairwise disjoint tubular neighborhoods modeled on real \(2\)-plane bundles over \(\RP^2\) whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than \(2\). When \(M\) is a homology \(4\)-sphere, the ambient error term vanishes, and the theorem recovers Massey's sharp inequality \(\abs{e(F)}\le 2g(F)\) for nonorientable surfaces in \(S^4\).

math.GT↗
arXiv2025

A triple torsion linking form and 3-manifolds in $S^4$

Given a rational homology 3-sphere $M$, we introduce a triple linking form on $H_1(M; \mathbb{Z})$, defined when the classical torsion linking pairing of three homology classes vanishes pairwise. If $M$ is the boundary of a simply-connected 4-manifold $N$, the triple linking form can be computed in terms of the higher order intersection form on $N$, introduced by Matsumoto. We use these methods to formulate an embedding obstruction for rational homology spheres in $S^4$, extending a 1938 theorem of Hantzsche.

math.GT↗
arXiv2024

Spineless 5-manifolds and the deformation conjecture

We construct a compact PL 5-manifold $M$ (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, $\vee^{}_{1 1}S^2$, which is "spineless", meaning $M$ is not the regular neighborhood of any 2-complex PL embedded in $M$. We formulate a related question about the existence of exotic smooth structures on 4-manifolds which is of interest in relation to the deformation conjecture for 2-complexes, also known as the generalized Andrews-Curtis conjecture.

math.GT↗
arXiv2022

The Geometry of the Bing Involution

In 1952 Bing published a wild (not topologically conjugate to smooth) involution $I$ of the 3-sphere $S^3$. But exactly how wild is it, analytically? We prove that any involution $I^h$, topologically conjugate to $I$, must have a nearly exponential modulus of continuity. Specifically, given any $\alpha>0$, there exists a sequence of $\delta$'s converging to zero, $\delta > 0$, and points $x,y \in S^3$ with dist$(x,y) < \delta$, yet dist$(I^h(x), I^h(y)) > \epsilon$, where $\delta^{-1} = e^{\left(\frac{\epsilon^{-1}}{\log^{(1+\alpha)}(\epsilon^{-1})}\right)}$, and dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ stretches distance much more than a Lipschitz function ($\delta^{-1} = c\epsilon^{-1}$) or a H\"{o}lder function ($\delta^{-1} = c^\prime(\epsilon^{-1})^{p}$, $1 < p < \infty$). Bing's original construction and known alternatives (see text) for $I$ have a modulus of continuity $\delta^{-1} > c \sqrt{2}^{\epsilon^{-1}}$, so the theorem is reasonably tight -- we prove the modulus must be at least exponential up to a polylog, whereas the truth may be fully exponential. Actually, the functional for $\delta^{-1}$ coming out of the proof can be chosen slightly closer to exponential than stated here (see Theorem 1). Using the same technique we analyze a large class of ``ramified'' Bing involutions and show, as a scholium, that given any function $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$, no matter how rapid its growth, we can find a corresponding involution $J$ of the 3-sphere such that any topological conjugate $J^h$ of $J$ must have a modulus of continuity $\delta^{-1}(\epsilon^{-1})$ growing faster than $f$ (near infinity). There is a literature on inherent differentiability (references in text) but as far as the authors know the subject of inherent modulus of continuity is new.

math.GT↗
arXiv2022

Shrinking Without Doing Much At All

In 1952 Bing astonished the mathematical world with his wild involution on $S^3$. It has been among the most seminal examples in topology. The example depends on finding shrinking homeomorphisms of Bing's decomposition of $S^3$ into points and arcs. If Bing's original homeomorphisms are varied, Bing's original wild involution changes by conjugation, which preserves some analytic properties \cite{fs22} while altering others. In 1988, Bing published a second paper "Shrinking Without Lengthening," answering a question that one of the present authors posed to him in an effort to understand the geometry of the entire conjugacy class. In this paper we produce a counterintuitive construction, namely, a method to shrink the Bing decomposition doing almost nothing at all--neither lengthening much nor rotating much.

math.GT↗
arXiv2022

The Smallest Interacting Universe

The co-emergence of locality between the Hamiltonian and initial state of the universe is studied in a simple toy model. We hypothesize a fundamental loss functional for the combined Hamiltonian and quantum state and minimize it by gradient descent. This minimization yields a tensor product structure simultaneously respected by both the Hamiltonian and the state, suggesting that locality can emerge by a process analogous to spontaneous symmetry breaking. We discuss the relevance of this program to the arrow of time problem. In our toy model, we interpret the emergence of a tensor factorization as the appearance of individual degrees of freedom within a previously undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the emergence of locality in Hamiltonians only, and found strong numerical confirmation of that raw Hilbert spaces of $\dim = n$ are unstable and prefer to settle on tensor factorization when $n=pq$ is not prime, and in [6] even primes were seen to "factor" after first shedding a small summand, e.g. $7=1+2\cdot 3$. This was found in the context of a rather general potential functional $F$ on the space of metrics $\{g_{ij}\}$ on $\mathfrak{su}(n)$, the Lie algebra of symmetries. This emergence of qunits through operator-level spontaneous symmetry breaking (SSB) may help us understand why the world seems to consist of myriad interacting degrees of freedom. But understanding why the universe has an initial Hamiltonian $H_0$ with a many-body structure is of limited conceptual value unless the initial state, $|ψ_0\rangle$, is also structured by this tensor decomposition. Here we adapt $F$ to become a functional on $\{g,|ψ_0\rangle\}=(\text{metrics})\times (\text{initial states})$, and find SSB now produces a conspiracy between $g$ and $|ψ_0\rangle$, where they simultaneously attain low entropy by settling on the same qubit decomposition.

hep-th↗
arXiv2021

The Universe from a Single Particle II

We continue to explore, in the context of a toy model, the hypothesis that the interacting universe we see around us could result from single particle (undergraduate) quantum mechanics via a novel spontaneous symmetry breaking (SSB) acting at the level of probability distributions on Hamiltonians (rather than on states as is familiar from both Ginzburg-Landau superconductivity and the Higgs mechanism). In an earlier paper [7] we saw qubit structure emerge spontaneously on $\mathbb{C}^4$ and $\mathbb{C}^8$, and in this work we see $\mathbb{C}^6$ spontaneously decomposing as $\mathbb{C}^2\otimes \mathbb{C}^3$ and very curiously $\mathbb{C}^5$ (and $\mathbb{C}^7$) splitting off one (one or three) directions and then factoring. This evidence provides additional support for the broad hypothesis: Nature will seek out tensor decompositions where none are present. We consider how this finding may form a basis for the origins of interaction and ask if it can be related to established foundational discussions such as string theory.

hep-th↗
arXiv2021

Building manifolds from quantum codes

We give a procedure for "reverse engineering" a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over $\mathbb{Z}$. Applying this procedure to chain complexes obtained by "lifting" recently developed quantum codes, which correspond to chain complexes over $\mathbb{Z}_2$, we construct the first examples of power law $\mathbb{Z}_2$ systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct.

math.DG↗
arXiv2021

PC4 at Age 40

This article is not a proof of the Poincaré conjecture but a discussion of the proof, its context, and some of the people who played a prominent role. It is a personal, anecdotal account. There may be omission or transpositions as these recollections are 40 years old and not supported by contemporaneous notes, but memories feel surprisingly fresh. I have not looked up old papers to check details of statements; this article is merely a download from my current mental state.

math.GT↗
arXiv2021

The Universe from a Single Particle

We explore the emergence of many-body physics from quantum mechanics via spontaneous symmetry breaking. To this end, we study potentials which are functionals on the space of Hamiltonians enjoying an unstable critical point corresponding to a random quantum mechanical system (the Gaussian unitary ensemble), but also less symmetrical local minima corresponding to interacting systems at the level of operators.

hep-th↗
arXiv2020

Filling links and spines in 3-manifolds

We introduce and study the notion of filling links in 3-manifolds: a link L is filling in M if for any 1-spine G of M which is disjoint from L, $\pi_1(G)$ injects into $\pi_1(M\smallsetminus L)$. A weaker "k-filling" version concerns injectivity modulo k-th term of the lower central series. For each k>1 we construct a k-filling link in the 3-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in 3-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable 3-manifold with $\pi_1(M)$ of rank 2.

math.GT↗
arXiv2020

Controlled Mather-Thurston theorems

Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor-Wood inequality is for circle bundles over surfaces, whereas the Mather-Thurston Theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern-Weyl theory and other smooth obstructions such as the Bott classes and the Godbillion-Vey invariant. Contradiction is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g. $\operatorname{PSL}(2,\mathbb{R})$ versus $\operatorname{U}(1)$ in the former case and $C^1$ versus $C^2$ in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather-Thurston cobordism to a semi-$s$-cobordism (ssc) and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we can't resolve the fine scale topology which may be present in the base, introduced by a ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, "distortions" of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell's $F \wedge F^\ast$ and Hilbert's $\int R\ dvol$ are replaced by an action which measures such "distortions." In this view, curvature results from renormalizing a discrete, group theoretic, structure.

math.GT↗
arXiv2020

Uniqueness in Haken's Theorem

Following Haken and Casson-Gordon, it was shown in [Sc] that given a reducing sphere or boundary-reducing disk E in a Heegaard split manifold M, the Heegaard surface T can be isotoped so that it intersects E in a single circle. Here we show that when this is achieved by two different positionings of T, one can be moved to the other by a sequence of 1) isotopies of T rel E 2) pushing a stabilizing pair of T through E and 3) eyegelass twists of T. The last move is inspired by one of Powell's proposed generators for the Goeritz group.

math.GT↗
arXiv2020

Predicting human-generated bitstreams using classical and quantum models

A school of thought contends that human decision making exhibits quantum-like logic. While it is not known whether the brain may indeed be driven by actual quantum mechanisms, some researchers suggest that the decision logic is phenomenologically non-classical. This paper develops and implements an empirical framework to explore this view. We emulate binary decision-making using low width, low depth, parameterized quantum circuits. Here, entanglement serves as a resource for pattern analysis in the context of a simple bit-prediction game. We evaluate a hybrid quantum-assisted machine learning strategy where quantum processing is used to detect correlations in the bitstreams while parameter updates and class inference are performed by classical post-processing of measurement results. Simulation results indicate that a family of two-qubit variational circuits is sufficient to achieve the same bit-prediction accuracy as the best traditional classical solution such as neural nets or logistic autoregression. Thus, short of establishing a provable "quantum advantage" in this simple scenario, we give evidence that the classical predictability analysis of a human-generated bitstream can be achieved by small quantum models.

quant-ph↗