Session 37. Wild algebraic and geometric topology

The obstruction to contractibility of Snake cones and Alternating cones

Andreas Zastrow, Institute of Mathematics, University of Gdansk, Poland
The talk is based on joint work with K. Eda, U. Karimov and D. Repovš
By Whitehead's Theorem it is clear that for CW-complexes any obstruction to contractibility will be in the homotopy groups. For general spaces this is naturally wrong. Also for Peano-continua there exists an infinite-dimensional counterexample by [2]. The question, whether also a finite-dimensional counter-example is possible, is still unanswered. The talk will discuss the construction techniques of some spaces which are somehow closest to providing corresponding counterexamples. The four construction principles (Abbr.: SC, CSC, AC, CAC) to be discussed turn a given space \(X\) into a space of one dimension higher in a way that the fundamental group becomes trivial, but the obstruction to contractibility may be expected to be retained. If applied to a one-dimensional manifold \(X\) a two-dimensional wild complex arises that has no closed surface as a subcomplex. However, in spite of this, \(\pi_2\) will be non-trivial. The talk (mainly presenting the results of [1], but also repeating the results of some predecessor-papers without the speaker's participation) will introduce and compare these functors SC, CSC, AC and CAC, explain why in general the results are not homotopy equivalent, but in spite of this \({ SC}(S^1) \simeq { CSC}(S^1) \simeq { AC}(S^1) \simeq { CAC}(S^1)\). In order to obtain this result it was necessary to compute the second homology group of a Hawaiian Earring-type product of the the torus-surface.
References
  1. K. Eda., U. H. Karimov,D. Repovš, A. Zastrow, On snake cones, alternating cones and related constructions , Glas. Mat. Ser. III 48(68) (2013), no. 1, 115-135.
  2. U. H. Karimov, D. Repovš, On noncontractible compacta with trivial homology and homotopy groups , Proc. Amer. Math. Soc. 138 (2010), no. 4, 1525-1531.
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