Cobordism properties of manifolds of small category

Mielke, M. V.

doi:10.1090/s0002-9939-1969-0236939-2

Cited by 3 publications

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“…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:…”

Section: Introductionmentioning

confidence: 99%

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Lusternik-Schnirelmann category and cobordism

Singh¹

1988

Proc. Amer. Math. Soc.

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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:

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Section: Introductionmentioning

confidence: 99%

“…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: THEOREM 2.12'.…”

Section: Introductionmentioning

confidence: 99%

Lusternik-Schnirelmann category and cobordism

Singh¹

1988

Proc. Amer. Math. Soc.

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Vector Bundles

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Cobordism classes of manifolds with category four

Singh¹

1988

Trans. Amer. Math. Soc.

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The Lusternik-Schnirelmann category of a manifold M is the smallest integer k such that M can be covered by fc 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.1The Lusternik-Schnirelmann category of M, cat(M), is the smallest integer k such that M can be covered by fc open sets each of which is contractible in M. In [3], Mielke showed that an n-dimensional manifold M with cat(M) < 3, and with ra = 3 (mod 4), is a boundary. The author, in [6], showed that one can determine the possible cobordism class of any manifold M with cat(M) < 3. The object of this paper is to determine the possible cobordism classes of manifolds M with cat(M) < 4.As in [6] the strategy is to use the fact that cat(M) is an upper bound for the length of nonzero products of the Stiefel-Whitney classes of M to determine the nonzero Stiefel-Whitney numbers, and hence the cobordism class of M. However, cat(JW) < 4 is a much weaker restriction than cat(M) < 3, and thus the classification problem is a lot harder.We begin §1 by recalling, from [6], the definition, and basic properties, of Poincare algebras and the cobordism category of a manifold M, denoted by cobcat(M). In §2 we begin the investigation by looking at even-dimensional manifolds, and show that an even-dimensional, nonbounding manifold M, with cobcat(M) < 4 and dim(M) ^ 2s + 2 for any s > 2, is cobordant to a square JV x N, where JV is also nonbounding, and cobcat(JV) < 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 ra, 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.

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Cobordism properties of manifolds of small category

Cited by 3 publications

References 3 publications

Lusternik-Schnirelmann category and cobordism

Lusternik-Schnirelmann category and cobordism

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Cobordism classes of manifolds with category four

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