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A note on generalized equivariant homotopy groups

Marek Golasiński, Daciberg L. Gonçalves, Peter N. Wong (2009)

Banach Center Publications

In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce Γ-Rhodes groups, where Γ admits a certain co-grouplike structure. Evaluation subgroups of Γ-Rhodes groups are discussed.

A Whitehead product for track groups (II).

Keith A. Hardie, A. V. Jansen (1989)

Publicacions Matemàtiques

Two direct relations are exhibited between the Whitehead product for track groups studied in [4] and the generalized Whitehead product in the sense of Arkowitz. The problem of determining the order of the Whitehead square is posed and some computations given.

Exploring W.G. Dwyer's tame homotopy theory.

Hans Scheerer, Daniel Tanré (1991)

Publicacions Matemàtiques

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,...

Homomorphic images and rationalizations based on the Eilenberg-MacLane spaces

Dae-Woong Lee (2005)

Czechoslovak Mathematical Journal

Are there any kinds of self maps on the loop structure whose induced homomorphic images are the Lie brackets in tensor algebra? We will give an answer to this question by defining a self map of Ω Σ K ( ℤ , 2 d ) , and then by computing efficiently some self maps. We also study the topological rationalization properties of the suspension of the Eilenberg-MacLane spaces. These results will be playing a powerful role in the computation of the same n -type problems and giving us an information about the rational homotopy...

Homotopy and homology groups of the n-dimensional Hawaiian earring

Katsuya Eda, Kazuhiro Kawamura (2000)

Fundamenta Mathematicae

For the n-dimensional Hawaiian earring ℍ n , n ≥ 2, π n ( ℍ n , o ) ≃ ℤ ω and π i ( ℍ n , o ) is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then H n ( X ∨ Y ) ≃ H n ( X ) ⊕ H n ( Y ) ⊕ H n ( C X ∨ C Y ) for n ≥ 1.

Loop spaces and homotopy operations

David Blanc (1997)

Fundamenta Mathematicae

We describe an obstruction theory for an H-space X to be a loop space, in terms of higher homotopy operations taking values in π * X . These depend on first algebraically “delooping” the Π-algebras π * X , using the H-space structure on X, and then trying to realize the delooped Π-algebra.

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