Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mapping class group of manifolds which look like $3$-dimensional complete intersections

Matthias Kreck, Yang Su

Abstract

In this paper we compute the mapping class group of closed simply-connected 6-manifolds $M$ which look like complete intersections, i.~e.~ $H_2(M;\mathbb Z) \cong \mathbb Z $ and $x^3 \ne 0$ where $x \in H^2(M; \mathbb Z)$ is a generator. We determine some algebraic properties of the mapping class group; for example we compute its abelianization and its center. We show that modulo the center the mapping class group is residually finite and virtually torsion-free. We also study low dimensional homology groups. The results are very similar to the computation of the mapping class group of Riemann surfaces. We give generators of the mapping class group, and generators and relations for the subgroup acting trivially on $π_{3}(M)$.

Mapping class group of manifolds which look like $3$-dimensional complete intersections

Abstract

In this paper we compute the mapping class group of closed simply-connected 6-manifolds which look like complete intersections, i.~e.~ and where is a generator. We determine some algebraic properties of the mapping class group; for example we compute its abelianization and its center. We show that modulo the center the mapping class group is residually finite and virtually torsion-free. We also study low dimensional homology groups. The results are very similar to the computation of the mapping class group of Riemann surfaces. We give generators of the mapping class group, and generators and relations for the subgroup acting trivially on .

Paper Structure

This paper contains 36 sections, 52 theorems, 233 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. The setting
  4. Computation of the mapping class group via the Torelli group
  5. Computation of the mapping class group via the special Torelli group
  6. Properties of the mapping class group
  7. Generators and presentations
  8. Dehn twists
  9. Generators
  10. Preliminaries
  11. Diffeomorphism classification
  12. The mapping class group of $\#\, r(S^{3} \times S^{3})$
  13. The surjectivity of $\rho$
  14. The Torelli group $\mathcal{I}(M)$
  15. Variation of the cup product and the Pontrjagin class
  16. ...and 21 more sections

Key Result

Theorem 2.3

Let $M$ be a manifold which looks like a $3$-dimensional complete intersection. Then there are short exact sequences The group $\mathcal{K}(M)$ only depends on whether $M$ is spin or not and $d(M)$ and $p(M)$. In Theorem thm:kernel we give tables of the groups $\mathcal{K}(M)$ in terms of $d(M)$ and $k(M)$. In the spin case it is isomorphic to a subgroup of $\mathbb Z/2 \times \mathbb Z/4 \times

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 76 more