Hans Scheerer

On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3∈R; let SRn denote the R-local n-sphere and define ΩRn:=SRn for n odd, ΩRn:=ΩΣSRn for n>0 even. An H-space (resp. a 1-conn. co-H-space) is “decomposable over R”, if it is homotopy equivalent to a weak product of spaces ΩRn (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M*(E;R); here M* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by π*(ΩC) as a cogroup in the category of M-Lie algebras. For R = Φ the functor [-,E] is also determined by the Lie algebra π*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.

Fibrewise construction applied to Lusternik-Schnirelmann category

In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted byQcat(X). It is obtained by applying a base-point free version ofQ=Ω∞∑∞ fibrewise to the Ganea fibrations. We provecat(X)≥Qcat(X)≥σcat(X) whereσcat(X) denotes Y. Rudyak’s strict category weight. However,Qcat(X) approximatescat(X) better, because, e.g., in the case of a rational spaceQcat(X)=cat(X) andσcat(X) equals the Toomer invariant.We show thatQcat(X×Y)≤Qcat(X)+Qcat(Y). The invariantQcat is designed to measure the failure of the formulacat(X×Sr)=cat(X)+1. In fact for 2-cell complexesQcat(X)<cat(X)⇔cat(X×Sr)=cat(X) for somer≥1.We note that the paper is written in the more general context of a functor λ from the category of spaces to itself satisfying certain conditions; λ=Q, ΩnΣn,Sp∞orLfare just particular cases.

Exploring W.G. Dwyer's tame homotopy theory

Let Sr be the category of r-reduced simplicial sets, r = 3; let Lr-1 be the category of (r-1)-reduced differential graded Lie algebras over Z. According to the fundamental work [3] of W.G. Dwyer both categories are endowed with closed model category structures such that the associated tame homotopy category of Sr is equivalent to the associated homotopy category of Lr-1. Here we embark on a study of this equivalence and its implications. In particular, we show how to compute homology, cohomology, homotopy with coefficients and Whitehead products (in the tame range) of a simplicial set out of the corresponding Lie algebra. Furthermore we give an application (suggested by E. Vogt) to p*(BG3) where BG3 denotes the classifying space of foliations of codimension 3.

The milnor-moore theorem in tame homotopy theory

LetX be a 1-connected space with Moore loop space ΩX. By a well-known theorem of J. W. Milnor and J. C. Moore [7] the Hurewicz homomorphism induces an isomorphism of Hopf algebrasU(π*(ΩX) ⊗Q)→H*(ΩX;Q). HereU(−) denotes the universal enveloping algebra and the Lie bracket on π*(ΩX) ⊗Q is given by the Samelson product.Assume now thatX is the geometric realization of anr-reduced simplicial set,r≥3. LetLX be a differential graded free Lie algebra over ℤ describing the tame homotopy type ofX according to the theory of [4]. Then the main result of the present paper is the construction of a sequence of morphisms of differential graded algebras betwenU(LX) and the algebraC U*(ΩX)z of normalized cubical chains on ΩX such that the induced morphisms on homology with coefficientsRk are isomorphismsHr-1+l(U(Lx);Rk) ≅Hr-1+lC U*(ΩX);Rk) forl≤k; hereR0⊆R1⊆… is a tame ring system, i. e.Rk)⊑Q and each primep with 2p−3≤k is invertible inRk.However, it is no longer true that the Pontrjagin algebraH≤r−1+k(ΩX; Rk) of ΩX in degrees ≤r−1+k is determined by π*(ΩX) or by a cofibrant (-fibrant) modelM of π*(ΩX) as will be shown by an example. But there is a filtration onH≤r−1+k(ΩX; Rk) such that the associated graded algebra is isomorphic toH≤r−1+k(U(M); Rk).This will be proved by using a filtered Lie algebra model ofX constructed from a bigraded model of π*(ΩX).

An application of algebraic R-local homotopy theory

Abstract Let R ⊆ Q be a subring, let r ≥ 3 and let m be an integer such that each prime p with 2p – 3 ≤ m – r is invertible in R. Assume that the r-reduced R-local CW-complex C has R-dimension ≤ m and is a co-H-space. Then C is homotopy equivalent to a wedge of Moore spaces. If H∗(C, R) is a free R-module, C is cogroup-like and r ≥ 4, then C is co-H-equivalent to a suspension.

Kaleidoscoping Lusternik-Schnirelmann category type invariants

A general LS-category type invariant is defined as a function of two variables. It specializes to (old and new) relative invariants and generates strong category notions. Each invariant is determined by an axiom scheme whose form has been established by Lusternik and Schnirelmann. We also discuss the corresponding absolute invariants and formulate, in a general category C, a method to define LS-category and strong LS-category concepts. They specialize to the usual ones and it turns out that some relative invariants are absolute invariants in the new sense.

LS-catégorie de CW-complexes à 3 cellules en théorie homotopique R-Locale

We study the Lusternik-Schnirelmann category of some CW-complexes with 3 cells, built on

A remark on a formula of Zassenhaus

Y=S^{2n}\cup_{k[\iota_{2n},\iota_{2n}]}e^{4n}

Groups of coalgebra morphisms and the Zassenhaus formulae

. In particular, we prove that an

Fibrations ` a la Ganea

R