(Math 592): Problem set 12

Bhargav Bhatt

1. Let G be a compact Lie group. Show that χ(G) 6= 0 if and only if G is a finite discrete group. 2. Say X is a finite CW complex. If a torus T := (S1)n acts continuously on X without fixed points, show that χ(X) = 0. (Hint: you may use that T is the closure of a subgroup gZ ⊂ T generated by a single g ∈ T .) 3. This exercise collects some basic properties of the Tor-functor. We write M for an . (a) Show that Tor(M, −) commutes with filtered direct limits. (b) Show that M is torsionfree if and only if Tor(M, −) = 0. (c) Let 0 → A → B → C → 0 be a SES of abelian groups. For any abelian group D, show that there is a LES 0 → Tor(A, D) → Tor(B,D) → Tor(C,D) → A ⊗ D → B ⊗ D → C ⊗ D → 0.

(d) Give an example showing that Tor(M, −) is not right exact. 4. Fix a commutative R and R-modules M, N. Choose a presentation 0 → K → P → M → 0 where P R Z is a free R-module and K is the kernel. Define Tor1 (M,N) := ker(K ⊗R N → P ⊗R N), so Tor1 (−, −) coincides with the functor Tor(−, −) used in class for abelian groups.

R (a) Show that Tor1 (M,N) is independent of the choice of the presentation. R (b) Show that Tor1 (M,N) is symmetric in M and N.

(c) For R = Z/4, give two complexes K and L of free R-modules such that H∗(K) and H∗(L) are bounded (i.e., live in only finitely many degrees), but H∗(K ⊗R L) is unbounded. Conclude that the algebraic R Kunneth¨ formula formulated in terms of the above Tor1 -functor (as discussed in class for R = Z) fails to describe H∗(K ⊗R L). (d) Show the analog of the LES in the previous problem holds true over R provided one drops the first 0. Give an example indicating why it is necessary to drop the first zero.

5. (Universal coefficient theorem) Let X be a space. Fix a positive n. Prove that there is a functorial (in X) short

0 → Hi(X)/n → Hi(X; Z/n) → Hi−1(X)[n] → 0.

Moreover, prove that this sequence is split (but not functorially, as we shall see in the next problem). 6. Fix positive m and n.

(a) Construct a CW complex X with a single cell in degrees 0, n and n + 1 such that Hei(X) = 0 for i 6= n and Hen(X) = Z/m. (b) By contemplating the map f : X → Sn+1 that collapses the n-skeleton of X, show that the splitting in the universal coefficient theorem is not functorial. (c) By contemplating the map (f, id) : X × X → Sn+1 × X, show that the splitting in the Kunneth formula is not functorial.

1 Please turn over... 7. Say X is a space and p is a prime. (a) Show that there is a short exact sequence of chain complexes

p 2 0 → C∗(X; Z/p) −→ C∗(X; Z/p ) → C∗(X; Z/p) → 0

where the first map is “mutiplication by p.” Taking the induced LES, we get a boundary map

i βp : Hi(X; Z/p) → Hi−1(X; Z/p)

which is called the Bockstein operator. Relate this operator to problem (3) above. i i+1 (b) Show that βp ◦ βp = 0 for all i. i (c) Show that βp = 0 if Hi−1(X) is p-torsionfree, and that the converse also holds true provided Hi−1(X) is finitely generated. Give an example indicating necessity of finite generation. i (d) Compute the Bockstein homomorphisms βp for the space X from the previous problem.

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