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Gluck twists on concordant or homotopic spheres

Daniel Kasprowski, Mark Powell, Arunima Ray

TL;DR

This work analyzes Gluck twists along embedded $2$-spheres in compact $4$-manifolds, comparing the twisted manifolds $M_S$ and $M_T$ when the spheres are concordant or homotopic. It proves that concordant spheres yield an $s$-cobordism between $M_S$ and $M_T$, which implies homeomorphism under suitable fundamental-group hypotheses; and that homotopic spheres yield simple homotopy equivalence, with additional surgery-theoretic conditions ensuring homeomorphism. The paper also presents counterexamples showing the necessity of the hypotheses and constructs explicit fake or exotic $4$-manifolds arising from Gluck twists, including cases where twists produce homeomorphic yet non-diffeomorphic or non-stably-diffeomorphic manifolds. The results extend to the topological category and illuminate when Gluck twists preserve or alter topology versus smooth structure in $4$-manifolds, highlighting the intricate interplay between concordance, homotopy, and surgery theory.

Abstract

Let $M$ be a compact $4$-manifold and let $S$ and $T$ be embedded $2$-spheres in $M$, both with trivial normal bundle. We write $M_S$ and $M_T$ for the $4$-manifolds obtained by the Gluck twist operation on $M$ along $S$ and $T$ respectively. We show that if $S$ and $T$ are concordant, then $M_S$ and $M_T$ are $s$-cobordant, and so if $π_1(M)$ is good, then $M_S$ and $M_T$ are homeomorphic. Similarly, if $S$ and $T$ are homotopic then we show that $M_S$ and $M_T$ are simple homotopy equivalent. Under some further assumptions, we deduce that $M_S$ and $M_T$ are homeomorphic. We show that additional assumptions are necessary by giving an example where $S$ and $T$ are homotopic but $M_S$ and $M_T$ are not homeomorphic. We also give an example where $S$ and $T$ are homotopic and $M_S$ and $M_T$ are homeomorphic but not diffeomorphic.

Gluck twists on concordant or homotopic spheres

TL;DR

This work analyzes Gluck twists along embedded -spheres in compact -manifolds, comparing the twisted manifolds and when the spheres are concordant or homotopic. It proves that concordant spheres yield an -cobordism between and , which implies homeomorphism under suitable fundamental-group hypotheses; and that homotopic spheres yield simple homotopy equivalence, with additional surgery-theoretic conditions ensuring homeomorphism. The paper also presents counterexamples showing the necessity of the hypotheses and constructs explicit fake or exotic -manifolds arising from Gluck twists, including cases where twists produce homeomorphic yet non-diffeomorphic or non-stably-diffeomorphic manifolds. The results extend to the topological category and illuminate when Gluck twists preserve or alter topology versus smooth structure in -manifolds, highlighting the intricate interplay between concordance, homotopy, and surgery theory.

Abstract

Let be a compact -manifold and let and be embedded -spheres in , both with trivial normal bundle. We write and for the -manifolds obtained by the Gluck twist operation on along and respectively. We show that if and are concordant, then and are -cobordant, and so if is good, then and are homeomorphic. Similarly, if and are homotopic then we show that and are simple homotopy equivalent. Under some further assumptions, we deduce that and are homeomorphic. We show that additional assumptions are necessary by giving an example where and are homotopic but and are not homeomorphic. We also give an example where and are homotopic and and are homeomorphic but not diffeomorphic.
Paper Structure (7 sections, 11 theorems, 16 equations, 1 figure, 1 table)

This paper contains 7 sections, 11 theorems, 16 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Effect on diffeomorphism type
  3. Conventions
  4. s-cobordism
  5. Simple homotopy equivalence
  6. Homeomorphism
  7. Fake and exotic manifolds from Gluck twists on homotopic spheres

Key Result

Theorem 1.1

The following statement holds in both the smooth category and the topological category with locally flat embeddings. Let $M$ be a compact $4$-manifold, and let $S,T\subseteq M$ be embedded, concordant $2$-spheres, both with trivial normal bundle. Then $M_S$ and $M_T$ are $s$-cobordant. If $\pi_1(M)$

Figures (1)

  • Figure 1: (a) A handle diagram for $(S^2 \times D^2) \# \mathbb{CP}^2$, relative to the boundary $S^2 \times S^1$. The $4$-handle is not shown. The $2$-handle corresponding to $\mathbb{CP}^1\subseteq \mathbb{CP}^2$ is shown in blue. (b) A handle diagram for the same manifold, but relative to a different parametrisation of the boundary, where the parametrisation has been altered by the Gluck twist. The grey arrow indicates a handle slide which produces the bottom figure. (c) The outcome of the handle slide shown in (b). The grey arrow indicates a further handle slide, which recovers the picture in (a).

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['thm:conc-gluck-thm']}
  • ...and 17 more