Monoids related to self homotopy equivalences of fibred product

TL;DR

This paper introduces a sequence of monoids related to self-homotopy equivalences of fibrewise pointed spaces, culminating in the group of such equivalences, and explores their properties and invariants.

Contribution

It defines a nested sequence of monoids for self-homotopy equivalences and studies their limits and invariants in the context of fibred products.

Findings

Defined a sequence of monoids converging to the group of self-homotopy equivalences.

Analyzed the monoid structure for fibred products in terms of component spaces.

Introduced invariants: self closeness number and self length.

Abstract

We introduce a nested sequence of monoids related to self-homotopy equivalences of fibrewise pointed spaces, such that the limit is the group of homotopy classes of fibrewise pointed self-equivalences. We explore this monoid for the fibred product in terms of individual spaces. Further we study two related invariants associated to these monoids: self closeness number and self length.