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.
