Simplicial resolutions and Ganea fibrations

TL;DR

This paper compares Ganea spaces and simplicial resolutions as approximations of path-connected spaces, constructing homotopy maps between them and exploring their relationships.

Contribution

It introduces maps between Ganea spaces and simplicial resolutions for simply connected spaces, and conjectures their existence for all levels.

Findings

Constructed maps between Ganea spaces and simplicial resolutions for simply connected spaces.

Proved the existence of a specific map for n=2 case.

Conjectured the general existence of such maps for all n.

Abstract

In this work, we compare the two approximations of a path-connected space $X$, by the Ganea spaces $G_n(X)$ and by the realizations $\|\Lambda_\bullet X\|_{n}$ of the truncated simplicial resolutions emerging from the loop-suspension cotriple $\Sigma\Omega$. For a simply connected space $X$, we construct maps $\|\Lambda_\bullet X\|_{n-1}\to G_n(X)\to \|\Lambda_\bullet X\|_{n}$ over $X$, up to homotopy. In the case $n=2$, we prove the existence of a map $G_2(X)\to\|\Lambda_\bullet X\|_{1}$ over $X$ (up to homotopy) and conjecture that this map exists for any $n$.