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Second homotopy classes associated with non-cancellative monoids

Kyoji Saito

Abstract

We construct second homotopy classes associated with twins of non-cancellative tuples of a monoid, where the monoid is defined by the semi-positive fundamental relations of the fundamental group of a CW-complex. As an application, we reconstruct the second homotopy classes for the complement of generic lines arrangement studied by Akio Hattori. We aim to apply the theory for the complement of elliptic discriminant loci in a forthcoming work.

Second homotopy classes associated with non-cancellative monoids

Abstract

We construct second homotopy classes associated with twins of non-cancellative tuples of a monoid, where the monoid is defined by the semi-positive fundamental relations of the fundamental group of a CW-complex. As an application, we reconstruct the second homotopy classes for the complement of generic lines arrangement studied by Akio Hattori. We aim to apply the theory for the complement of elliptic discriminant loci in a forthcoming work.
Paper Structure (2 sections, 4 theorems, 45 equations)

This paper contains 2 sections, 4 theorems, 45 equations.

Table of Contents

  1. Example of Hattori $\pi_2$-class
  2. Appendix: Classifying spaces of monoids

Key Result

Proposition 4.6

1. The actions of division inertia groups $\tilde{a}\backslash [\tilde{a}\tilde{b}\tilde{d},\tilde{a}\tilde{b}\tilde{d}]_{\mathcal{R}_g}/\tilde{d}$ and $\tilde{a}\backslash [\tilde{a}\tilde{c}\tilde{d},\tilde{a}\tilde{c}\tilde{d}]_{\mathcal{R}_g}/\tilde{d}$ on $[\tilde{b},\tilde{c}]$ from left and 2. Let $\tilde{a}_i,\tilde{b}_i,\tilde{c}_i, \tilde{d}_i \!\in \! \mathcal{F}^+$ ($i\!=\!1,2$) be r

Theorems & Definitions (10)

  • Proposition 4.6
  • proof
  • Corollary 4.7
  • Remark 4.8
  • Definition 4.9
  • Proposition 4.10
  • proof
  • Definition 4.11
  • Proposition 4.12
  • proof