On parallelizability and span of the Dold manifolds

Abstract

The Dold manifold P (m, n) is obtained from the product S m × CP n of the m-dimensional sphere and n-dimensional complex projective space by identifying (x, [z 1 , . . . , z n+1 ]) with (−x, [z 1 , . . . ,z n+1 ]), wherez denotes the complex conjugate of z. We answer the parallelizability question for the Dold manifolds P (m, n) and, by completing an earlier ( 2008) result due to Peter Novotný, we solve the vector field problem for the manifolds P (m, 1).

On generalized Dold manifolds

“…The case when the flag manifold is a complex projective space corresponds to the classical Dold manifold P (m, n − 1). In this special case the above result is due to J. Korbaš [9]. See also [22] in which J. Ucci characterized classical Dold manifolds which admit codimension-one embeddings in the Euclidean space.…”

“…The classical Dold manifold corresponds to r = 2 and n 1 ≥ n 2 = 1. Theorem 1.1 in this special case is due to J. Korbaš [9]. (Cf.…”

“…Let P (m, n) denote the space obtained as the quotient by the cyclic group Z 2 -action on the product S m × CP n generated by the involution (u, L) → (−u, L), u ∈ S m , L ∈ CP n where L denotes the complex conjugation. The spaces P (m, n), which seem to have first appeared in the work of Wu, are called Dold manifolds, after it was shown by Dold [6] that, for suitable values of m, n, the cobordism classes of P (m, n) serve as generators in odd degrees for the unoriented cobordism algebra N. Dold manifolds have been extensively studied and have received renewed attention in recent years; see [9], [15] and also [14], [21], and [4].…”

“…Acknowldegments: Sankaran thanks Peter Zvengrowski for bringing to his attention the papers of Július Korbaš [9] and Peter Novotný [15].…”

On generalized projective product spaces and Dold manifolds

“…[13] and [7] for few of them. Recently, in [14], Nath and Sankaran make a slight generalization of Dold manifolds.…”