On smooth manifolds with homotopy type of a homology sphere

TL;DR

This paper investigates the classification of smooth manifolds with the same simple homotopy type as a given space, focusing on the normal bundles they admit, especially for certain homology spheres.

Contribution

It introduces a map from the set of manifolds to a quotient of K-theory and characterizes its image for specific homology spheres, revealing new conditions for normal bundle realizations.

Findings

Determined the image of the map for certain homology spheres.

Identified conditions under which elements of K(X) are pullbacks of normal bundles.

Provided criteria for manifolds to have the same simple homotopy type with specified normal bundles.

Abstract

In this paper we study $\mathcal M(X)$, the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of $X$, via a map $\Psi$ from $\mathcal M(X)$ into the quotient of $K(X)=[X,BSO]$ by the action of the group of homotopy classes of simple self equivalences of $X$. The map $\Psi$ describes which bundles over $X$ can occur as normal bundles of manifolds in $\mathcal M(X)$. We determine the image of $\Psi$ when $X$ belongs to a certain class of homology spheres. In particular, we find conditions on elements of $K(X)$ that guarantee they are pullbacks of normal bundles of manifolds in $\mathcal M(X)$.