On the ku-homology of certain classifying spaces II (original) (raw)

On the ku-homology of certain classifying spaces

2010

We calculate the ku-homology of the groups Z/p^n X Z/p and Z/p^2 X Z/p^2. We prove that for this kind of groups the ku-homology contains all the complex bordism information. We construct a set of generators of the annihilator of the ku-toral class. These elements also generates the annihilator of the BP-toral class.

Topological K–(co)homology of classifying spaces of discrete groups

Algebraic & Geometric Topology, 2013

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction EG × G X of a proper G-CW -complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology K * (BG) and K * (BG) up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where G is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.

Rational computations of the topological K-theory of classifying spaces of discrete groups

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.

Homotopy colimits of classifying spaces of abelian subgroups of a finite group

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q, G), q ≥ 2, using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces B(q, G) p ⊂ B(q, G) defined for a fixed prime p. Then B(q, G) is stably homotopy equivalent to a wedge of B(q, G) p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial 2-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups of order 2 2n+1 , n ≥ 2, B(2, G) does not have the homotopy type of a K(π, 1) space. For a finite group G, we compute the complex K-theory of B(2, G) modulo torsion.

On the bordism ring of complex projective space

Proceedings of the American Mathematical Society, 1973

The bordism ring MUt{CPa>) is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings Et(CPm) are considered, where E is an oriented ring spectrum, R=7rt(£), andpR=0 for a prime/». It is known that Et(CPcc) is freely generated as an .R-module by elements {ßT\r^0}. The ring structure, however, is not known. It is shown that the elements {/VI^O} form a simple system of generators for £t(CP°°) and that ßlr=s"rß"r mod(/?j, • • • , ßvr-i) for an element s e R (which corresponds to [CP"-1] when E=MUZV). This may lead to information concerning Et(K(Z, n)).

Some conditions on the homology groups of the Koszul complex

Pacific Journal of Mathematics, 1984

In this paper we introduce the concept of a (d, i)-sequence (d, i E N) in a commutative ring A, noetherian and with identity (cf. Def. 1.1). Let K(z, A) be the Koszul complex on A, with respect to the sequence z = Z],... ,z n : the concept of a (d, z)-sequence is expressed in terms of the structure of HχK(z, A)); in particular, it turns out that z is an (n, z)-sequence iff HχK(z, A)) = 0, and such a condition implies z is a (d, /)-sequence for any d < n. If z u ... ,z Λ is a (d, /)-sequence in h A = A/(z h+] ,... ,z n),d<h < «, then z is seen_to be a (d, /)-sequence in A so, in particular, if HχK(z; d A))-0 in d A 9 then z is a (d, /)-sequence. Moreover, for i = 1, the two conditions are equivalent, so that z is a (d, \)-sequence means precisely that z u ...,z d \s regular in d A. For i> 1, examples show that z is a (d, i)-sequence is a condition strictly weaker than Zj,... ,z Λ is a (d, i)-sequence in h A, and we investigate the relationship between those two properties. In fact, their equivalence allows us to read the depth of a quotient ring A/(z h+ ,,... ,z n) in terms of the Koszul complex K(z; A) and implies, for (d, /)-sequences, properties which are a natural generalization of good properties satisfied by regular sequences, such as the depth-sensitivity of the Koszul complex. A characteristic condition for their equivalence is a kind of weak surjectivity of a natural map acting between syz /+1 (^(z; A)) and syz /+1 (#(z; h A)). From an algebraic form of that weak surjectivity we get some sufficient conditions, in terms of weak regularity of the sequence z h+ι ,...,z n. For instance, if z Λ+1> ... ,z n is a ^/-sequence, or a relative regular sequence, or less, if z h+ ,,... ,z w is a relative regular A-sequence with respect to a convenient set of ideals, then zisa(d, i)-sequence in A implies z u. ..,z h is a(d, i)-sequence in h A. Moreover, if z is a (d, z)-sequence and z d+ι ,...,z n is a regular sequence, then H^Kiz; A)) = 0, while this vanishing implies that it is possible to find x ι9 ...,x n in /=(ZJ,...,Z B) such that Zj,...,^], x l9 ...,x n isa(d, z)-sequence and x d+] >... ,x n is a regular sequence. In the last section we give an interpretation of our results in terms of the behaviour of some systems of linear equations. N. 1. Let A be a noetherian ring (with 1) and z = z,,...,z ll a sequence of elements of A such that (z l9 ...,z n)A ^A. We denote by K(z\ A) the Koszul complex with respect to z, i.e. the differential graded algebra (DGA for short) (cf. [G-L] cap. I for a definition) 0 137 138 CARLA MASSAZA AND ALFIO RAGUSA generated by e o i-1,...,«, with differential d J (e Jχ Λ Ae tj) = ^ 2 (~ l) r+1^ e^ Λ Λ^ Λ Λe z/ Also we write syz ι (K(z\ A)) = ker(d I _ 1) C Λ'^U" for / = 1,... ,n + 1. As in [M-R], for every 1 < / < rf < π, Γ/ π '^ will mean the free ^4-module generated by e Jχ .. Ί-e Λ Λe y , with 1 <j x <-< j\ < n and j) > d, which is a complementary module of Λ'(Ae ι θ ••• θ^e^), briefly ΛU,... έ/ ,in A ι A n 9 so Λ ι A n = ΛUp.^θ Then χ z : Λ ^4"-» Λ ι A λ ... d will be the usual projections, i.e. χ t \ ?, a e, ... I = Λ ^h and, more generally, χf:ΛU^Λ'4.. Λ (A</i) will be like χ z-when we set J = Λ. When z l9 ... ,z n are fixed elements of ^4, we write t A=A/(z t+ι ,...,z n)A for ί = 0,... 9 n (n A = A), and z i E t A means the image of z, by the natural map ρ t \ A-* t A.

Some Properties of Homology Groups of Khalimsky Spaces

2015

In this paper we introduce the digital singular homology groups of the digital spaces topologized by the Khalimsky topology by constructing the digital standard n-simplexes. Then we'll compute the digital singular homolo gy groups of some basic digital spaces up to the dimension 2 and investigate that the digital singul ar homology theory for the digital spaces is a functor from the category KDTC of KD-topological category to the category Ab of abelian groups.