Eilenberg-Mac Lane Complex Research Articles

Rational Reidemeister trace of an outer automorphism of finite order

Let H be a group of type FP over the rationals and let ϕ:H→H be an automorphism whose class in Out(H) has finite order and such that the Lefschetz number of some iterate ϕi is nonzero. We show that under a suitable Bass conjecture assumption, this iterate ϕi is conjugate to a finite order automorphism, and the intersection of the fixed point set of ϕ with the center of H is of bounded exponent. In the torsion-free case it follows that the center of the HNN extension 〈H,t|∀h∈H:tht−1=ϕ(h)〉 is infinite cyclic. These results are related to Geoghegan and Nicas' first order Euler characteristic of a group G with a finite Eilenberg-MacLane complex; this is a morphism χ1(G) from the center of G to H1(G;k) where k is a ring of coefficients. We give a complete determination of χ1(G) with rational coefficients.

The group of self-homotopy equivalences of the m-fold smash product of a space

Let E(X) be the set of homotopy classes of self-homotopy equivalences of a space X. The set E(X) is a group by composition of homotopy classes. We study the group E(X∧m) for the m-fold smash product X∧m. We show that the two obvious homomorphisms φ:Sm→E(X∧m) and ψ:E(X)m→E(X∧m) define a homomorphism Ψ:E(X)m⋊Sm→E(X∧m) for any space X, where E(X)m⋊Sm is the semi-direct product of the product group E(X)m by the symmetric group Sm. We show that in most cases the homomorphism φ:Sm→E(X∧m) is a monomorphism and the kernel of Ψ is isomorphic to the kernel of ψ. The injectivity of Ψ is established for the complex projective n-space CPn(n≥2), and hence, E((CPn)∧m) contains a subgroup isomorphic to {±1}m⋊Sm. Sufficient conditions for Ψ to be injective are obtained for the Eilenberg–MacLane complex K(Ar,n) where A is a subring of Q or a ring Z/k(k≥2) and Ar is the free A-module of rank r. From this result, we see that E(K(Ar,n)∧m) contains a subgroup isomorphic to GLr(A)m⋊Sm in many cases.

Resolutions for metrizable compacta in extension theory

We prove a K-resolution theorem for simply connected CW-complexes K in extension theory in the class of metrizable compacta X. This means that if K is a connected CW-complex, G is an abelian group, n ∈ N≥2, G = π n (K), π k (K) = 0 for 0 ≤ k 2. Thus, in case K is an Eilenberg-Mac Lane complex of type K(G, n), then (c) becomes dime Z < n. If in addition π n+1 (K) = 0, then (a) can be replaced by the stronger statement, (aa) π is K-acyclic. To say that a map π is K-acyclic means that for each x ∈ X, every map of the fiber π -1 (x) to K is nullhomotopic.

The product formula for Lusternik–Schnirelmann category

If C=C_\phi denotes the mapping cone of an essential phantom map \phi from the suspension of the Eilenberg-Mac Lane complex K=K(Z,5) to the 4-sphere S=S^4 we derive the following properties: (1) The LS category of the product of C with any n-sphere S^n is equal to 3; (2) The LS category of the product of C with itself is equal to 3, hence is strictly less than twice the LS category of C. These properties came to light in the course of an unsuccessful attempt to find, for each positive integer m, an example of a pair of 1-connected CW-complexes of finite type in the same Mislin (localization) genus with LS categories m and 2m. If \phi is such that its p-localizations are inessential for all primes p, then by the main result of [J. Roitberg, The Lusternik-Schnirelmann category of certain infinite CW-complexes, Topology 39 (2000), 95-101], the pair C_*, C where C_*= S wedge \Sigma ^2 K, provides such an example in the case m=1.

THE EILENBERG-BORSUK THEOREM FOR MAPPINGS INTO AN ARBITRARY COMPLEX

The classical Eilenberg-Borsuk theorem on extension of partial mappings into a sphere is generalized to the case of an arbitrary complex . It is formulated in terms of extraordinary dimension theory, which is developed in the present paper. When is an Eilenberg-MacLane complex, the result can be expressed in terms of cohomological dimension theory. For partial mappings of an -manifold , the following is obtained: Theorem. If , then there exists a compactum of dimension , such that the mapping extends to and for every abelian group with the cohomological dimension of with coefficients in does not exceed . Thus, in comparison with the classical Eilenberg-Borsuk theorem, there is obtained an additional condition as to the cohomological dimension of . Bibliography: 17 titles.

Minimal 4-manifolds for groups of cohomological dimension 2

We show that if π is a group with a finite 2-dimensional Eilenberg-Mac Lane complex then the minimum of the Euler characteristics of closed 4-manifolds with fundamental group π is 2χ(K(π, 1)). If moreoverMis such a manifold realizing this minimum then π2(M) ≅Similarly, if π is a PD3-group andw1(M) is the canonical orientation character of π then χ(M)≧l and π2(M) is stably isomorphic to the augmentation ideal ofZ[π].

Infinite dimensional compacta having cohomological dimension two: an application of the Sullivan conjecture

IN [S], A. N. Dranishnikov produced examples of infinite dimensional metric compacta that have integral cohomological dimension equal to three. These examples established the distinct nature of these classical dimension theories, settling the problem, posed by P. S. Alexandroff in 1932 [l], of whether the integral cohomological dimension of a compact metric space is the same as its covering dimension. The techniques used in [S] shed little light on whether or not there could be such an example having integral cohomological dimension two. The infinite dimensionality of the examples in [S] is, ultimately, detected using the vanishing of the reduced complex K-theory with Z/p coefficients (p a prime) of an Eilenberg-MacLane complex K(Z, 3). While reduced complex K-theory with Z/p coefficients of an Eilenberg-MacLane complex K(Z, n) vanishes for n 2 3, the same is not true for K(Z, 2)). (The reader is referred to [2] and [4] for details of these K-theoretic assertions.) The specific nature of K-theory itself plays no direct role in the analyses in [S]. The essential feature is that it is a generalized cohomology theory for which K(Z, 3) behaves as a point; i.e., the reduced K-theory of K(Z, 3) is trivial. Actually, the latter is not true for complex K-theory itself but is valid when Z/p coefficients (p a prime) are used (see [2], [4]). The absence of readily available generalized cohomology theory for which K(Z, 2) is known to behave as a point requires an alternate approach to that in [S] in order to produce an infinite dimensional compact metric space having integral cohomological dimension equal to two. (In addition, the generalized cohomology theory would have to have the property that, in each dimension, its value on a finite complex is a finite group, the latter is true for K-theory with jinite coeficients.) An overview of the approach providing an alternate method to that in [S] and producing cohomological dimension two examples is:

On the spaces of self-homotopy equivalences for fibre spaces

If a base space B is simply connected, we had some results on G0(E) in the previous papers [16, 17, 18, 19]. Here we treat G0(£) for the case of a nonsimply connected base space B. Let G be an abelian group and let Aut(G) be its group of automorphisms. Denote by L(G, n+1) the classifying space for fibrations with fibre K(G, ri) and by W an Eilenberg-MacLane complex K(Aut(G), 1). Then we have the fibration :

On the spaces of self homotopy equivalences for fibre spaces

If a base space B is simply connected, we had some results on G0(E) in the previous papers [16, 17, 18, 19]. Here we treat G0(£) for the case of a nonsimply connected base space B. Let G be an abelian group and let Aut(G) be its group of automorphisms. Denote by L(G, n+1) the classifying space for fibrations with fibre K(G, ri) and by W an Eilenberg-MacLane complex K(Aut(G), 1). Then we have the fibration :

Rational homotopy of the space of sections of a nilpotent bundle

We show that an algebraic construction proposed by Sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle. In his paper Uhomologie des espaces fonctionnels, R. Thom studied the homotopy type of the space F* of continuous maps of X into F homotopic to a given map/. Starting from a Postnikov decomposition of F, he built the functional space F* step by step. He also indicated how one could construct a differential graded algebra describing the rational homotopy type of F*. Later on, Sullivan gave an algebraic model which mirrors this construction in terms of a DG-algebra representing X and the minimal model of F. The aim of this paper is to show, following the method of Thom, that the model of Sullivan is indeed a model for the functional space under suitable restrictions. As in (3), we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections Ts of a nilpotent fiber space p: Y -» X homotopic to a given section s. In §1 we explain Thorn's geometric construction. In §2 we describe an algebraic model for an abelian Galois covering of a nilpotent space. In §3 we show how the model of Sullivan fits with the geometry. I thank the referee for many improvements of this paper. 1. A Postnikov factorization of the space of sections. 1.1. Let G be a finitely generated abehan group and let X be a path connected space whose cohomology groups Hk(X G) are finitely generated for each k. To avoid difficulties with the topologies (cf. (4)), we can work in the category of simplicial sets. Proposition (Thom (4)). The space K(G, m)x of continuous maps of X in the Eilenberg-Mac Lane complex K(G, m) is homotopically equivalent to the product nr=rA °fthe Eilenberg-Mac Lane spaces Ki = K(Hm-'(X; G), i). More precisely, let x G Hm(K(G; m); G) be the fundamental class of K(G; m). If e:K(G,m)X XX^K(G,m)

An example for homotopy groups with coefficients

In this note we present two spaces X and Y all of whose homotopy groups are isomorphic, but whose homotopy groups with coefficients are not isomorphic for a certain coefficient group.' This example depends on the fact that the universal coefficient sequence [2], which relates the ordinary homotopy groups to those with coefficients, does not split. The spaces X and Y will be 1-connected, CW-complexes and the coefficient group will be Zm, the integers modulo m. We adopt the following notation: M(G, p) denotes a Moore complex of type (G, p) (i.e., a space with a single nonvanishing homology group G in dimension p) and K(G, p) denotes an Eilenberg-MacLane complex of type (G, p) (i.e., a space with a single nonvanishing homotopy group G in dimension p). Recall that 7rr(G; A), the rth homotopy group of the space A with coefficients in the group G, is the group of homotopy classes of base point preserving maps from M(G, r) into A. If Z is the group of integers, then 7r(Z; A) =irr(A), the rth homotopy group of A. Finally we recall the universal coefficient theorem [2] which asserts the exactness of the following sequence

Some Additive Cohomology Operations which are not Suspensions

If X is a space with base point, let ΩX denote the space of loops of X based at its base point. We will denote the homology and cohomology suspensions by σ* and σ*, respectively, , . Let K(π, n) be the Eilenberg-MacLane complex with only one non-vanishing homotopy group πn(K(π, n))= π, πi(K(π, n)) = 0 for i ≠ n. If π is abelian K(π, n) = ΩK(π, n+1). We will assume that π is a finitely generated abelian group throughout this note. Let K(π, n) × K(π, n) → K(π, n) denote the multiplication in K(π, n).