Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Morse theory for group presentations

Ximena Fernández

Abstract

We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews--Curtis conjecture do satisfy the conjecture.

Morse theory for group presentations

Abstract

We introduce a novel combinatorial method to study -transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplified Morse CW-complex, with an explicit description of the attaching maps and bounds on the dimension of the complexes involved in the deformation. We apply this technique to show that some known potential counterexamples to the Andrews--Curtis conjecture do satisfy the conjecture.

Paper Structure

This paper contains 6 sections, 13 theorems, 41 equations, 12 figures.

Table of Contents

  1. Discrete Morse theory and Whitehead deformations
  2. $Q^{**}$-transformations of group presentations
  3. Applications to the Andrews--Curtis conjecture
  4. Experimental results.
  5. Computational procedure for examples in Section \ref{['examples']}
  6. The Morse presentation $Q_{K'_{{\mathcal{G}}_{2k-1}}, M_k}$

Key Result

Theorem \oldthetheorem

MR1612391MR1939695 Let $K$ be a regular CW-complex and let $f:K\to {\mathbb{R}}$ be a discrete Morse function. Let $a < b$ be real numbers.

Figures (12)

  • Figure 1: CW-regular structure of the torus $T$. Arrows (in white) indicate the orientation of cells.
  • Figure 2: Left: Face poset associated to the regular CW-structure of the torus $T$, an acyclic matching $M$ in red and labels corresponding to an induced discrete Morse function $f_M$ (critical cells are the empty bullets). Right: Regular CW-structure of the torus $T$. The arrows in red indicate pairs of cells in the matching $M$.
  • Figure 3: Internal collapses in the torus $T$. The CW-complex obtained after performing successively the internal collapses associated to the matched pairs 9 (a), 8 (b), 7 (c), 5 (d), 3 (e) and 2 (f).
  • Figure 4: Top: Two subdivided solid pyramids $K_1$ and $K_2$ with the vertices $v_i$ and edges $x_i$ identified according to the labels and arrows. The 2-cells in the boundary of $K_1$ and $K_2$ are also identified following the identification of their faces. Bottom: The top view of $K_1$ and $K_2$.
  • Figure 5: Acyclic matching $M$ in ${\mathcal{X}}(K)$ in solid red (and a maximal acyclic matching $M_{\max}$ in solid red and black). In dashed lines, the subposets ${\mathcal{X}}(K_1)$, ${\mathcal{X}}(K_2)$ and ${\mathcal{X}}(P)$ with $P$ the CW-structure $P$ projective plane in the basis of $K_1$ and $K_2$.
  • ...and 7 more figures

Theorems & Definitions (38)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 28 more