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Universität Augsburg, SS 2012 Prof. Dr. K. Cieliebak, Evgeny Volkov String Topology Problem Set 1. Problem 1 Suppose that X is a CW complex and e ∈ X a 0-cell. Show that every 4 Points continuous map µ : X × X → X such that the maps x 7→ µ(x, e) and x 7→ µ(e, x) are homotopic to the identity through maps X → X (not necessarily preserving e) is homotopic to a map µ e : X × X → X such that µ e(x, e) = µ e(e, x) = x for all x ∈ X (in particular, µ e defines an H-space structure). Conclude that every homotopy equivalence X → Y between an H-space X and a CW complex Y induces an H-space structure on Y. Problem 2 Every H-space X is abelian, i.e., π1 X is abelian and acts trivially on each 4 Points πk X, k ∈ N. Problem 3 If X is an H-space, then for any pointed space K the set < K, X > of 4 Points base point preserving homotopy classes of maps K → X inherits a product with unit. If X = ΩY , then < K, ΩY > is a group. What is this group for K = pt and for K = S n ? Problem 4 Show that any Eilenberg-MacLane space K(n, Z) has a unique H-space 4 Points structure up to homotopy. Hint: use that for any space X the set of homotopy classes of maps < X, K(n, Z) > is in one to one correspondence with the set of cohomology classes H n (X, Z). Problem 5∗ Let X and Y be H-spaces. Show that X×Y is an H-space and the Pontrjagin product on its homology (with coefficients in a field) is given by H∗ (X × Y ) ⊗ H∗ (X × Y ) −→ H∗ (X × Y ), 0 (a × b) · (a0 × b0 ) 7→ (−1)|a ||b| (a · a0 ) × (b · b0 ). Problems marked with a star are more difficult than the others, but (we think) we know how to solve them. We do not know how to solve problems marked with double star. Solutions can be handed in on Wednesday 24.10.12 at the beginning of the exercise class