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Non-cobordant hyperbolic manifolds

Jacopo G. Chen

TL;DR

The paper proves that for every $n\ge 4$ with $n\not\equiv 3\pmod{4}$ there exists a connected closed hyperbolic $n$-manifold that does not bound a compact $(n+1)$-manifold, by linking the unoriented cobordism class to the fixed-point set of an involution and employing a geodesic embedding of arithmetic hyperbolic manifolds. Central to the method is expressing $[M]$ in terms of projective bundles over fixed submanifolds and computing the Thom homomorphism value $\varphi^n[M]$ via Stiefel–Whitney data; this yields a computable invariant that distinguishes cobordism classes. The Kolpakov–Reid–Slavich embedding is extended to manifolds with involutions and, together with a twist operation, enables iterative dimension-raising while preserving arithmetical structure to control the invariant. The paper also outlines strategies to handle the remaining dimensions $4m+3$ (excluding $n=2^k-1$) and discusses the implications for even dimensions, including explicit starting examples and the limitations of the invariant in fully determining cobordism classes. Overall, the work provides a constructive framework that realizes nontrivial elements in the unoriented cobordism ring by hyperbolic geometry and offers concrete starting points and inductive tools for broad dimensional coverage.

Abstract

In all dimensions $n \ge 4$ not of the form $4m+3$, we show that there exists a closed hyperbolic $n$-manifold which is not the boundary of a compact $(n+1)$-manifold. The proof relies on the relationship between the cobordism class and the fixed point set of an involution on the manifold, together with a geodesic embedding of Kolpakov, Reid and Slavich. We also outline a possible approach to cover the dimensions $4m+3 \ne 2^k-1$.

Non-cobordant hyperbolic manifolds

TL;DR

The paper proves that for every with there exists a connected closed hyperbolic -manifold that does not bound a compact -manifold, by linking the unoriented cobordism class to the fixed-point set of an involution and employing a geodesic embedding of arithmetic hyperbolic manifolds. Central to the method is expressing in terms of projective bundles over fixed submanifolds and computing the Thom homomorphism value via Stiefel–Whitney data; this yields a computable invariant that distinguishes cobordism classes. The Kolpakov–Reid–Slavich embedding is extended to manifolds with involutions and, together with a twist operation, enables iterative dimension-raising while preserving arithmetical structure to control the invariant. The paper also outlines strategies to handle the remaining dimensions (excluding ) and discusses the implications for even dimensions, including explicit starting examples and the limitations of the invariant in fully determining cobordism classes. Overall, the work provides a constructive framework that realizes nontrivial elements in the unoriented cobordism ring by hyperbolic geometry and offers concrete starting points and inductive tools for broad dimensional coverage.

Abstract

In all dimensions not of the form , we show that there exists a closed hyperbolic -manifold which is not the boundary of a compact -manifold. The proof relies on the relationship between the cobordism class and the fixed point set of an involution on the manifold, together with a geodesic embedding of Kolpakov, Reid and Slavich. We also outline a possible approach to cover the dimensions .
Paper Structure (12 sections, 13 theorems, 34 equations, 1 figure, 2 tables)

This paper contains 12 sections, 13 theorems, 34 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The unoriented cobordism ring
  3. Decomposable classes
  4. Projective bundles and involutions
  5. Fixed points of involutions
  6. Applying phin
  7. Geodesic embeddings
  8. Twisting the embedding
  9. Non-cobordant hyperbolic manifolds
  10. A starting manifold
  11. The remaining dimensions
  12. Even dimensions

Key Result

Theorem 1.1

For each $n \ge 4$, $n \not \equiv 3 \pmod 4$, there exists a connected, non-cobordant closed hyperbolic $n$-manifold.

Figures (1)

  • Figure 4.1: Construction of the twist $T(M', \tau', M, \tau)$ when $M$ does not separate $M'$. Clockwise from top left: the manifold $M$ embedded in $M'$; the double cover $N$ with deck automorphism $s$; the twisted quotient $M"$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2: thom
  • Theorem 2.3: dold
  • Definition 2.4
  • Theorem 3.1: diff-periodic, borel-hirzebruch
  • Theorem 3.2: diff-periodic
  • Theorem 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 16 more