Bordism Classification Of Manifolds With Commuting Involutions

This dissertation consists of three chapters.In the first chapter, we discuss the involutions fixing the disjoint union of product of real projective space RP(3) with the quaternionic projective space HP(k).Let (M, T) be a smooth closed manifold with a smooth involution T:M→Mwhose fixed point set is F. For F=(?)RP_i(3)×HP_i(k), we show the existence of involutions and prove that every involution bounds by constructing ingeniously symmetric polynomial and computing characteristic number.In the second chapter, we discuss the involutions fixing (?)CP_i(1)×HP_i(n), where CP(1) and HP(n) denote the 1-dimensional complex projective space and n-dimensional quaternionic projective space respectively,Let (M, T) be a smooth closed manifold with a smooth involution T whose fixed point set is F=(?)CP_i(1)×HP_i(n). In this chapter, we show the existence of involutions fixing(?)CP_i(1)×HP_i(n) and prove that every involution bounds by constructing ingeniouslysymmetric polynomial and computing characteristic number.In the third chapter, we discuss the problem of commuting involutions with fixed point set RP(2m+1)∪RP(2n+1).LetΦ:(Z₂)²Ã—M→M be a smooth action of the group (Z₂)²={T₁,T₂|T_i²= 1,T₁T₂=T₂T₁} on a smooth closed manifold M. Let T₃=T₁T₂. The fixed point data ofΦis (F_Φ;ε₁,ε₂,ε₃), where F_Φ={x∈M|T_i(x)=x, i=1,2,3} is a closed manifold,ε_i is the normal bundle of F_Φin F_Ti={x∈M|T_i(x)=x},i=1, 2,3.In this chapter, we prove the following: Let (M,Φ) be a smooth (Z₂)²-action on a closed and smooth manifold M whose fixed point set is F_Φ=RP(2m+1)∪RP(2n + 1), where RP(2m+1) and RP(2n+1) are projective spaces with dimensions 2m + 1 and 2n + 1 respectively. Letbe the fixed point data ofΦ. If at least twoμ_i's have dimension greater than 2m + 1, and at least one v_i has dimension greater than 2n + 1, where m≥n, then (M,Φ) bounds equivariantly.