Self-closeness numbers of finite cell complexes

TL;DR

This paper investigates the properties of self-closeness numbers in finite cell complexes, establishing inequalities, relations with connectivity, and calculating specific examples, thereby advancing understanding of their algebraic and topological characteristics.

Contribution

It introduces new inequalities and relations involving self-closeness numbers, computes these numbers for key examples, and explores their properties using Sullivan and Quillen models.

Findings

Self-closeness number is bounded above by homology dimension.

Relations between self-closeness numbers and manifold connectivity are established.

Explicit calculations of self-closeness numbers for projective and lens spaces.

Abstract

We reformulate the inequalities among self-closeness numbers of spaces in cofibrations making use of homology dimension and show that the self-closeness number of a space is less than or equal to the homology dimension of the space. Then we prove a relation of self-closeness numbers and the connectivity for manifolds satisfying Poincar\'{e} duality. On the other hand we determine the self-closeness numbers of the real projective spaces, lens spaces and a cell complex defined by Mimura and Toda. Moreover, making use of the models of Sullivan and Quillen, we show several properties of self-closeness number for finite cell complexes, and rational examples are udied to obtain some precise results. Finally, we prove relations among self-closeness numbers defined by homotopy groups, homology groups and cohomology groups.