Distributions, vector distributions, and immersions of manifolds in Euclidean spaces11The author was supported in part by two grants of VEGA (Slovakia).

Differential Topology of Semimetals

“…Atiyah-Thomas Z 2 -invariant. A detailed study of tangent 2-fields with finite singularities was carried out in [63,64,2,30], where analogues of the Poincaré-Hopf theorem for vector fields were obtained. For this, we need the notion of the local index at a singularity w ∈ W of a tangent 2-field (a, b) on T \ W .…”

“…2-plane fields are also called oriented 2-distributions, see[30] for a guide to k-distributions on manifolds.…”

On parallelizability and span of the Dold manifolds

“…By the Hopf theorem on vector fields, a smooth closed connected manifold has span at least one if and only if its Euler-Poincaré characteristic vanishes. For further information, we refer to [7], [8], [9], [17].…”

“…In addition to the span of a manifold M , one defines its stable span ( [7], [8], [9]), denoted stable span(M ), to be the integer span(T M ⊕ ε) − 1. Note that the geometric dimension of a real vector bundle α is defined to be the smallest g such that there exists a g-dimensional real vector bundle which is stably isomorphic to α (recall that two vector bundles are stably isomorphic if their sums with suitable trivial bundles are isomorphic).…”

“…Note that the geometric dimension of a real vector bundle α is defined to be the smallest g such that there exists a g-dimensional real vector bundle which is stably isomorphic to α (recall that two vector bundles are stably isomorphic if their sums with suitable trivial bundles are isomorphic). In this terminology, stable span(M q ) = k means that the geometric dimension of the (tangent bundle of the) manifold M q is q − k. For the question of how stable span and span are related, as well as for several topics where the knowledge of these invariants can be useful, see, for instance, [7].…”