Download String Topology Problem Set 1.

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