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Self homotopy equivalences of classifying spaces of compact connected Lie groups

Stefan Jackowski, James McClure, Bob Oliver (1995)

Fundamenta Mathematicae

We describe, for any compact connected Lie group G and any prime p, the monoid of self maps B G p → B G p which are rational equivalences. Here, B G p denotes the p-adic completion of the classifying space of G. Among other things, we show that two such maps are homotopic if and only if they induce the same homomorphism in rational cohomology, if and only if their restrictions to the classifying space of the maximal torus of G are homotopic.

Spaces associated to quadratic endofunctors of the category of groups.

Hans-Joachim Baues, Teimuraz Pirashvili (2005)

Extracta Mathematicae

Square groups are gadgets classifying quadratic endofunctors of the category of groups. Applying such a functor to the Kan simplicial loop group of the 2-dimensional sphere, one obtains a one-connected three-type. We consider the problem of characterization of those three-types X which can be obtained in this way. We solve this problem in some cases, including the case when π2(X) is a finitely generated abelian group. The corresponding stable problem is solved completely.

Strong surjectivity of mappings of some 3-complexes into 3-manifolds

Claudemir Aniz (2006)

Fundamenta Mathematicae

Let K be a CW-complex of dimension 3 such that H³(K;ℤ) = 0, and M a closed manifold of dimension 3 with a base point a ∈ M. We study the problem of existence of a map f: K → M which is strongly surjective, i.e. such that MR[f,a] ≠ 0. In particular if M = S¹ × S² we show that there is no f: K → S¹ × S² which is strongly surjective. On the other hand, for M the non-orientable S¹-bundle over S² there exists a complex K and f: K → M such that MR[f,a] ≠ 0.

Strong surjectivity of mappings of some 3-complexes into M Q 8

Claudemir Aniz (2008)

Open Mathematics

Let K be a CW-complex of dimension 3 such that H 3(K;ℤ) = 0 and M Q 8 the orbit space of the 3-sphere 𝕊 3 with respect to the action of the quaternion group Q 8 determined by the inclusion Q 8 ⊆ 𝕊 3 . Given a point a ∈ M Q 8 , we show that there is no map f:K → M Q 8 which is strongly surjective, i.e., such that MR[f,a]=min(g −1(a))|g ∈ [f] ≠ 0.

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