Dr. Severin Bunk Algebra und Zahlentheorie Fachbereich Mathematik Algebraic Topology Universit¨atHamburg Summer 2018

Sheet 8 Solutions are due on 08.06.18.

Problem 8.1 (a) Show that for each abelian group D, the tensor product ( ) D is right-exact. That is, α β − ⊗ show that for every short exact sequence 0 A B C 0 of abelian groups, there → → → → is an exact sequence of abelian groups

α id β idD A D ⊗ D B D ⊗ C D 0 . ⊗ ⊗ ⊗

β (b) Prove that if 0 A α B C 0 is split exact, then for any abelian group D the → → → → induced sequence

α id β idD 0 A D ⊗ D B D ⊗ C D 0 . ⊗ ⊗ ⊗ is exact.

(c) Is the abelian group (Q, +) free? Show that ( ) Q is exact. − ⊗ Hint: Observe that for any abelian group A and field k, the tensor product A k is a ⊗ k-vector space. For the second part, start by working out the kernel of the map qA : A A Q, a a 1. → ⊗ 7→ ⊗ (d) The rank of an abelian group A can be defined as

rk(A) := dim (A Q) . Q ⊗ n Show that if A is finitely generated, i.e. A = Z Zq1 Zqm , then rk(A) = n. Moreover, α β ∼ ⊕ ⊕· · ·⊕ show that if 0 A B C 0 is any short exact sequence of finitely generated abelian → → → → groups, then rk(B) = rk(A) + rk(C) .

1 Problem 8.2 Let X be a a finite CW complex. The Euler characteristic of X is defined as

n χ(X) := ( 1) rk Hn(X, Z) Z . − ∈ n N0 X∈
(a) Let cn N0 denote the number of n-cells in X for n N Show that ∈ ∈ χ(X) = ( 1)n c . − n n N0 X∈

(b)A triangulation of a topological space X is a simplicial complex KX together with a homeomorphism KX ∼= X. Given the fact that the compact 2-dimensional surface Fg of 1 genus g N0 admits a (highly non-unique) triangulation, prove Euler’s famous formula ∈ E K + F = 2 2g , − − where E is the number of vertices (German: “Ecken”), K is the number of edges (Ger- man: “Kanten”), and F is the number of faces (German: “Fl¨achen”) in any arbitrary

triangulation of Fg. (c) Let X,Y be finite CW complexes. Compute χ(X Y ). t (d) Let X = X X be a finite CW complex that can be written as the union of two 1 ∪ 2 subcomplexes. Express the Euler characteristic χ(X) in terms of the Euler characteristics χ(X ), χ(X ), and χ(X X ). 1 2 1 ∩ 2

n n n+1 n Problem 8.3 Let f : S S be a continuous map of degree m Z, and let X := D f S → ∈ ∪ be the topological space obtained from attaching an (n + 1)-cell to the n-sphere Sn along f. Compute the homology of X with values in an arbitrary abelian group G.

1According to the German Wikipedia, this formula appeared first, though not in this generality, in 1758!

2