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The Lusternik-Schnirelmann category of a manifold M is the smallest integer k such that M can be covered by k open sets each of which is contractible in M. The classification up to cobordism of manifolds with category 3 was completed by the author in 1985. The object of this paper is to attempt a similar classification of manifolds with category 4. Introduction. Let M be an n-dimensional manifold.' The LusternikSchnirelmann 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. In [3], Mielke showed that an n-dimensional manifold M with cat(M) 2, is cobordant to a square N x N, where N is also nonbounding, and cobcat(N) < 4. This along with other results reduces the problem to one of looking at the odd dimensions. We look at the odd dimensions in ?3, and show that for many such dimensions n, with more than three terms in their 2-adic expansions, there is no nonbounding n-dimensional manifold M with cat(M) < 4. Computations in small dinmensions suggest conjectures in the other dimensions. However, there does not seem to be a general way of handling these dimensions. Finally, in ?4, we give some applications to finding lower bounds for the category of spin manifolds, and also show that a spin manifold M, with cat(M) < 4, is nonoriented cobordant to the square of an oriented manifold. Received by the editors July 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R75, 55M30.

Cobordism classes of manifolds with category four