Homology of Cell Complexes

Cooke, George E.; Finney, Ross Lee

doi:10.1515/9781400877751

Cited by 63 publications

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“…To (1). Let x"* (0) be an arbitrary element of niE[S(i)' Since S(i) is additively closed, Therefore, one only need to show that there is a bound for the ith components of elements of G \ Si' For this purpose, choose an element z E S \ S(i)' Let H denote the additive group generated by S(i) and z in zn.…”

Section: Proof Note That Ho(dj = K[si] and Hp(djmentioning

confidence: 99%

Affine semigroups and Cohen-Macaulay rings generated by monomials

Trung¹,

Hoa²

1986

Trans. Amer. Math. Soc.

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Section: Proof Note That Ho(dj = K[si] and Hp(djmentioning

confidence: 99%

Affine semigroups and Cohen-Macaulay rings generated by monomials

Trung¹,

Hoa²

1986

Trans. Amer. Math. Soc.

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“…The problem is that even though the individual pieces are homeomorphic to balls and are glued together nicely, the boundaries of the closures of the pieces are not homeomorphic to spheres in general. (If they were, then the Voronoǐ decomposition would give rise to a regular cell complex [CF67], which can be used as a substitute for a simplicial or CW complex in homology computations.) Nevertheless, there is a way to remedy this.…”

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confidence: 99%

Modular Forms, a Computational Approach

Stein

2007

Graduate Studies in Mathematics

264

289

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“…This generalizes readily to permutation representations. The notation for joins used here will be that of Cooke and Finney [4]. Specifically, Cooke and Finney define a raised chain complex of a cell complex E to be the chain complex sC, where sC'(E) = Cq_x(E), if q > 0, and sC0(E) = ZvE, where vE is a void cell of dimension -1 added to E. The differentials are given by d'q = dq_x if a > 0, and 90 = e, the augmentation of C. They then show that the chain complex of the join E * E' of two cell complexes £ and £' is given by: sC*(E * £') = sC+(E) ® ¿C*(£').…”

Section: He\e a Point Of E) = [He\e A Cell Of E)mentioning

confidence: 99%

The Classifying Space of a Permutation Representation

Blowers¹

1977

Transactions of the American Mathematical Society

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Abstract. In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.I. Introduction. In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. It will be shown that the cohomology groups of this classifying space (with coefficients in Z) are equal to the corresponding cohomology groups of the permutation representation. (The cohomology theory of permutation representations can be found in Snapper [9] and Harris [6].) The standard example of a classifying space of a group will be generalized to a classifying space of a permutation representation.In a previous paper of the author [1], it was proved that the cohomology of the join of two permutation representations (to be defined later) has trivial cup-products. That paper noted that the algebraic, computational proof of this result was lengthy. In this paper it will be shown that a classifying space for the join of two permutation representations is given by the topological join of their classifying spaces, which will lead to a brief topological proof of this result.

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Homology of Cell Complexes

Cited by 63 publications

References 0 publications

Affine semigroups and Cohen-Macaulay rings generated by monomials

Affine semigroups and Cohen-Macaulay rings generated by monomials

Modular Forms, a Computational Approach

The Classifying Space of a Permutation Representation

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