Problem Set 4
Due Thursday, April 21
Problem 1. Let G be a topological group and φ : G → G an inner automorphism of G,
i.e., φ is conjugation by some element of G. Show that Bφ : BG → BG is homotopic to the
identity.
Problem 2. Let n ≥ 2 and let d be a nonzero integer. Let fd : S n → S n be a map of degree
d (i.e., a map inducing multiplication by d on Hn (S n )), and let Fd be the homotopy fiber of
fd .
(a) Using the Serre spectral sequence, compute the homology of Fd .
(b) Determine the structure of the Serre spectral sequence, starting from E 2 , for the homotopy fiber sequence ΩS n → Fd → S n . Ignore signs.
Hint: Recall that
(
Z if i is a multiple of n − 1,
Hi (ΩS n ) =
0 otherwise.
(c) Assume n ≥ 3. Determine the structure of the Serre spectral sequence, starting from
E 2 , for the homotopy fiber sequence ΩS n → ΩS n → Fd . Determine, up to signs, the
group extensions needed to assemble H∗ (ΩS n ) from the E ∞ page.
Problem 3.
(a) Pretend that you don’t know K(Z, 2) ' CP ∞ . Use the Serre spectral sequence for
S 1 → ∗ → K(Z, 2) to compute the cohomology ring H ∗ (K(Z, 2)).
(b) Use the Serre spectral sequence for K(Z, 2) → ∗ → K(Z, 3) to compute the ring
H ∗ (K(Z, 3))/H ≥14 (K(Z, 3)).
Problem 4. Let n ≥ 1 be odd. Show that the cohomology ring H ∗ (ΩS n+1 ) has the form
ΛZ (γ1 ) ⊗ ΓZ (γ2 ) for some cohomology classes γ1 in degree n and γ2 in degree 2n. Explicitly,
H ∗ (ΩS n+1 ) ∼
= Z[γ1 , γ2k | k ≥ 1]/(γ 2 , γ k − k!γ2k ).
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Problem 5. Let L and M be complex line bundles over a CW complex X. Show that
c1 (L ⊗ M ) = c1 (L) + c1 (M ) in H 2 (X).
Hint: The universal pair of line bundles lives over CP ∞ × CP ∞ .
Problem 6. Let E be a finite-dimensional (real or complex) vector space and let n ≥ 0.
(a) Let γn (E) = {(V, x) ∈ Grn (E)×E |x ∈ V }. Show that the projection γn (E) → Grn (E)
is a vector bundle. This is called the tautological bundle over Grn (E).
(b) Let Aff n (E) be the set of n-dimensional affine subspaces of E, i.e., translates of ndimensional linear subspaces. Put a topology on this set and show that the “translate
to the origin” map Aff n (E) → Grn (E) is a vector bundle.
(c) Explain why the tangent bundle of Grn (E) is Hom(γn (E), Aff n (E)).
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