Abstract :

In this talk, I will describe the notion of generalized Dold manifold and stable parallellelizability and cobordism classes of it. Let $(X,J)$ be a compact connected almost complex manifold. A smooth involution $\sigma : X \to X$ is called a complex conjugation if the differential $T\sigma : TX \to TX$ anti-commutes with $J$. We assume that $\fix(\sigma)$ is non-empty. Then its dimension is equal to one-half of the real dimension of $X$. The involution $\theta$ on $\mathbb{S}^m \times X$ defined as $(v,x) \mapsto (-v,\sigma(x))$ is a fixed point free involution and the quotient space $P(m,X)$ obtained from $\mathbb{S}^m \times X$ by the identification $(v,x) \sim (-v, \sigma(x))$ is a smooth manifold which we called a generalized Dold manifold. The classical Dold manifold corresponds to the case when $X$ is the complex projective space with $\sigma$ being the usual complex conjugation. We describe the tangent bundle of $P(m,X)$, and, under a mild restriction we obtain a formula for the Stiefel-Whitney classes of $P(m,X)$. We obtain applications to the stable parallellelizability and cobordism classes of $P(m,X)$ especially when $X$ is a complex flag manifold. This is a joint work with Parameswaran Sankaran.