Topology of generalized Dold manifolds[HBNI Th161]

Abstract

Classical Dold manifolds were defined as the orbit space of Z 2 action on the product of a sphere and a complex projective space where Z 2 acts on the sphere by antipodal involution and the complex projective space by complex conjugation. Dold has given the description of Z 2 - cohomology ring of Dold manifolds. He also obtained the formula for Stiefel- Whitney polynomial of Dold manifolds. He used these manifolds to obtain generators for unoriened cobordism ring in odd dimesions. Ucci obtained the formula for stable tangent bundle of Dold manifolds. Korbaˇs obtained the criterion for parallelizability and stable parallelizability of Dold manifolds. In this thesis, we obtain a generalization of the Dold manifolds where we replace the complex projective space by an almost complex manifold admitting a complex conjugation. We call them as generalized Dold manifolds. We obtain a description of the tangent bundle, under a mild hypothesis a formula for Stiefel-Whitney polynomial of generalized Dold manifolds, a criteria for orientability and spin structures as an applications of simple compuations of first and second Stiefel-Whitney classes. Using the description of tangent bundle, we also obtain estimates for span and stable span of generalized Dold manifolds. We obtain a very general criterion for (non) vanishing of cobordism classes of generalized dold manifolds. We applied our results by taking almost complex manifolds as complex Grassmann manifolds, more generally as complex flag manifolds. Our proof to determine estimates for span and stable span of generalized Dold manifolds involves Bredon-Kosi ́ nski’s theorem, certain functor introduced by Lam to study the immersions of flag manifolds. We apply Stefiel-Whitney numbers argument to determine the (non) vanishing of cobordism classes of generalized Dold manifolds. We also use the theory of Clifford algebras, a result of Conner and Floyd concerning cobordism of manifolds admitting stationary point free action of elementary abelian 2-group actions to obtain the results of (non) vanishing of cobordism classes of eneralized Dold manifolds corresponding to Grassmann manifolds.