Loop homological invariants associated to real projective spaces

TL;DR

This paper derives a formula for the mod 2 reduced Poincaré series of a specific loop space involving real projective spaces, under certain homological conditions on a subspace.

Contribution

It provides a new explicit formula for the mod 2 reduced Poincaré series of a loop space constructed from real projective spaces and a subspace, expanding understanding of their homological invariants.

Findings

Derived a formula for the mod 2 reduced Poincaré series of the loop space.

Established conditions under which the formula applies.

Enhanced the understanding of homological invariants related to real projective spaces.

Abstract

Let A be a based subspace of Y. Under the assumptions that Y is path-connected and that the reduced diagonal map of A induces the zero map in all mod 2 reduced homology groups, we compute a formula for the mod 2 reduced Poincar\'{e} series of the loop space $\Omega ((A \wedge \mathbb{RP}^\infty) \cup_{A \wedge \mathbb{RP}^1} (Y \wedge \mathbb{RP}^1))$. Here $\mathbb{RP}^\infty$ and $\mathbb{RP}^1$ denote the infinite real projective space and the real projective line respectively.