On generalized projective product spaces and Dold manifolds

TL;DR

This paper introduces a broad generalization of projective product spaces and Dold manifolds, exploring their topological properties and computing their homology, cohomology, tangent bundles, and vector fields.

Contribution

It presents a unified framework for generalized projective product spaces and Dold manifolds, leading to new classes of smooth manifolds and detailed topological analyses.

Findings

Computed integral homology groups and cohomology rings.

Analyzed stable tangent bundles and vector field problems.

Established properties of the new manifold classes.

Abstract

D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the generators of the non-oriented cobordism ring. Recently, in 2019, A. Nath and P. Sankaran made a modest generalization of Dold manifolds. In this paper we simultaneously generalize both the notions of projective product spaces and Dold manifolds, leading to infinitely many different classes of new smooth manifolds. Our main goal will be to study the integral homology groups. cohomology rings, stable tangent bundles, and vector field problems, on certain generalized projective product spaces and Dold manifolds.