Algebraic topology (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 2 T .) 3. This exercise collects some basic properties of the Tor-functor. We write M for an abelian group. (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 ring 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 integer n. Prove that there is a functorial (in X) short exact sequence 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 integers 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. 2.