Lusternik-Schnirelmann category and cobordism

Abstract

ABSTRACT. The Lusternik-Schnirelmann category of a manifold M is the smallest integer k that M can be covered by k open sets each of which is contractible in M. It is an upper bound for the length of nonzero products of Stiefel-Whitney classes of M. The object of this paper is to use this restriction, on the length of nonzero products, to investigate the cobordism classes of manifolds with category less than or equal to three. Introduction.Let M be an n-dimensional manifold.1 The Lusternik-Schnirelmann category of M, cat(M), is the smallest integer k such that M can be covered by k open sets each of which is contractible in M. The object of this paper is to identify the cobordism class of a manifold M with cat(M) < 3. Thom [11] has shown that the Stiefel-Whitney numbers (see [7]) of M determine the cobordism class of M. It is also known (see [4]) that cat(M) is greater than the length of the longest nonzero product in H*(M)-the coefficient ring will always be Z2. In particular if cat(M) < 3, then the only possibly nonzero Stiefel-Whitney numbers are W,W"_¿[M] for 0 < i < n. Mielke [5] has shown that if n = 3 (mod4), then all these numbers are zero and thus M is a boundary. We will see in §2 that this is a special case of the following:

Cobcat and singular bordism

“…Analogous to [1], given a singular manifold (M n , /) in X there is associated a Poincare algebra P* given as follows:…”

“…Let P* be the Poincare algebra associated to the singular manifold (M n , /) in X. As in [1], an element z of any graded algebra A* will be called ^-decomposable if it is zero or is the sum of the products Z\-Z2 z p where z z e A* with dim z f > 0 for each /, and p > k. From now on, the ambient space X will be taken to be the universal base space BO and {M n , f) will denote a singular manifold in BO with cobcat(M", /) < 3. Proof, (a) By the hypothesis, the last proposition and the fact that cobcat(Λ/ n , /) < 3, we have …”

Cobordism classes of manifolds with category four

“…We begin by recalling the concepts of cobordism category, introduced in [6], and Poincare algebras [9].…”

“…Wy = 0 in p*, and so by Proposition 2.8, Wj = 0 in p* for all odd j. Thus M is cobordant to a square N x N (see[4,6]), with W^, ••■IVip[JV] = W2l, ■ ■ ■ W2ip [M] for any partition ii H-hip = n/2. Thus clearly JV is nonbounding, and cobcat(JV) < 4.…”