The Vector Field Problem for Homogeneous Spaces

Abstract

Dedicated to Professor Peter Zvengrowski with admiration and respect.Abstract. Let M be a smooth connected manifold of dimension n ≥ 1. A vector field on M is an association p → v(p) of a tangent vector v(p) ∈ T p M for each p ∈ M which varies continuously with p. In more technical language it is a (continuous) cross-section of the tangent bundle τ (M ). The vector field problem asks: Given M , what is the largest possible number r such that there exist vector fields v 1 , . . . , v r which are everywhere linearly independent, that is, v 1 (x), . . . , v r (x) ∈ T x M are linearly independent for every x ∈ M . The number r is called the span of M , written span(M ). It is clear that 0 ≤ span(M ) ≤ dim(M ). The vector field problem is an important and classical problem in differential topology. In this survey we shall consider the vector field problem focussing mainly on the class of compact homogeneous spaces.2010 Mathematics Subject Classification. 57R25.

On generalized projective product spaces and Dold manifolds

“…Thus sp(P (M, N )) ≥ 1 if and only if one of X (M ), X (N ) is zero, by [15,Theorem 1.7]. Since sp(E) ≥ sp(B) for a smooth fiber bundle F ֒→ E → B, then we have sp(P (M, N )) ≥ sp(M/Z 2 ).…”

“…These manifolds were introduced to study the generators of the non-oriented cobordism ring [5]. From then on several interesting properties of Dold manifolds are studies, see [8], [15],…”

“…In general, this problems are open even on product of spheres. Reader are refer to, [8] and [15] for some motivational background on the vector field problems. We note that if S m admits an r-field, it also admits a linear r-field which is equivariant with respect to antipodal action.…”