Bott and Tu Reading Notes

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Applications of Spectral Sequences

Notation and Remarks

  • For \(M\) a manifold, \(T(M)\) is the unit tangent bundle of \(M\)
  • For \(R\) a ring \(R\delta_i\) denotes a copy of \(R\) appearing in the \(i\)th (co)homological degree
  • \(S^n \subset {\mathbb{R}}^{n+1}\) and \(S^{2n-1} \subset {\mathbb{C}}^n\)
  • Theorem: \(F \to E \to B\) a fibration results in \(E_2^{p,q} = H^p(B, H^q(F; G)) = H^p(B;G) \otimes H^q(F; G)\) for nice enough spaces \(X\) and groups \(G\)
    • Corollary: \(H^n(X\times Y) = \displaystyle\bigoplus_{p+q=n} H^p(X, H^q(Y))\)
  • Facts about tensor products
    • \((rm)\otimes n = r(m\otimes n) =m \otimes(rn)\)
    • \((r+s)(m\otimes n) = rm\otimes n + sm\otimes n\)
    • \({\mathbb{Z}}_p \otimes_{\mathbb{Z}}{\mathbb{Z}}_q = {\mathbb{Z}}/\gcd(p,q)\) and \(\gcd(p,q) = 1\) yields 0.
    • Some computations:
      • \({\mathbb{Z}}_n \otimes_{\mathbb{Z}}{\mathbb{Q}}= 0\)
      • \({\mathbb{Z}}_n \otimes_{\mathbb{Z}}{\mathbb{Q}}/{\mathbb{Z}}= 0\)
      • \({\mathbb{Q}}\otimes_{\mathbb{Z}}{\mathbb{Q}}= {\mathbb{Q}}\)
      • \(({\mathbb{Q}}/{\mathbb{Z}})\otimes_{\mathbb{Z}}{\mathbb{Q}}= 0\)
      • \({\mathbb{Q}}/{\mathbb{Z}}\otimes_{\mathbb{Z}}{\mathbb{Q}}/{\mathbb{Z}}= 0\)
      • \(R[x]\otimes_R S \cong S[x]\)
      • \(k \to K\) a field extension: \(k[x]/(f) \otimes_k K \cong K[x]/(f)\)
    • Symmetric, Associative
    • \((\oplus A_i )\otimes B = \oplus(A_i \otimes B)\)
    • \({\mathbb{Z}}\otimes A = A\)
    • \({\mathbb{Z}}_n \otimes A = \frac{A}{nA}\)

List of Results

  • A simply connected \(n\)-dimensional manifold \(M_n\) is orientable
    • Use \(S^{n-1} \to T(M_n) \to M_n\)
  • \(H^*({\mathbb{CP}}^2) = {\mathbb{R}}\delta_0 + {\mathbb{R}}\delta_2 + {\mathbb{R}}\delta_4\)
    • Use \(S^1 \to S^5 \to{\mathbb{CP}}^2\)
  • \(H^*({\mathbb{CP}}^2) = \frac{{\mathbb{R}}[x]}{(x^3)}\)
    • Use \(S^1 \to S^5 \to{\mathbb{CP}}^2\)
  • \(H^*({\mathbb{CP}}^n) = \displaystyle\sum_{i=0}^n{\mathbb{R}}\delta_{2i}\)
    • Use \(S^1 \to S^{2n+1} \to{\mathbb{CP}}^n\)
  • \(H^*({\mathbb{CP}}^n) = \frac{{\mathbb{R}}[x]}{(x^{n+1})}\)
    • Use \(S^1 \to S^{2n+1} \to{\mathbb{CP}}^n\)
  • \(H^*(SO^3) = {\mathbb{Z}}\delta_0 + {\mathbb{Z}}_2\delta_2 + {\mathbb{Z}}\delta_3\)
    • Use \(S^1 \to T(S^2) \to S^2\) and identify \(T(S^2) = SO^3\)
    • Also use \(E_2^{p,q} = H^p(S^2) \otimes H^q(S^1)\)
  • \(H^*(SO^4) = ?\)
    • Use \(SO^3 \to SO^4 \to S^3\)
  • \(H^*(U^n) = ?\)
    • Use \(U^{n-1} \to U^n \to S^{2n-1}\)
  • \(H^*(\Omega S^2) = \displaystyle\sum_{i=0}^\infty {\mathbb{Z}}\delta_i\)
    • Use \(\Omega S^2 \to PS^2 \to S^2\)
    • Also use \(E_2^{p,q} = H^p(S^2, H^q(\Omega S^2))\)
  • \(H^*(\Omega S^3) = \displaystyle\sum_{i=0}^\infty {\mathbb{Z}}\delta_{2i}\)
    • Use \(\Omega S^3 \to PS^3 \to S^3\)
  • \(H^*(\Omega S^n) = \displaystyle\sum_{i=0}^\infty {\mathbb{Z}}\delta_{i(n-1)}\)
    • Use \(\Omega S^3 \to PS^3 \to S^3\)
  • \(H^*(\Omega S^2) = \frac{{\mathbb{Z}}[x]}{(x^2)} \otimes{\mathbb{Z}}\left\{{1,e, \frac{1}{2!}e^2,\cdots}\right\}, \dim x = 1, \dim e = 2\)
    • Use \(\Omega S^3 \to PS^3 \to S^3\)
  • \(H^*(\Omega S^n) = \frac{{\mathbb{Z}}[x]}{(x^2)} \otimes{\mathbb{Z}}\left\{{1,e, \frac{1}{2!}e^2,\cdots}\right\}, \dim x = n-1, \dim e = 2(n-1_)\)
    • Use \(\Omega S^3 \to PS^3 \to S^3\)

List of Fibrations

  • \(S^1 \to S^{2n+1} \to{\mathbb{CP}}^n\), the Hopf fibration?

  • \(S^3 \to S^{4n+3} \to\mathbb{HP}^n\) the generalized Hopf fibration? (not used here)

  • Hopf Fibrations

    • \(S^0 \to S^1 \to S^1\)
      • Induced by $S^1 
        \subset 
        {\mathbb{R}}
        ^2 
        \to
        S^1 = 
        {\mathbb{R}} 
        \cup 
        \infty
        $
    • \(S^1 \to S^3 \to S^2\)
      • Induced by \(S^3 \subset {\mathbb{C}}^2 \to S^2 = {\mathbb{C}}\cup\infty\)
    • \(S^3 \to S^7 \to S^4\)
      • Induced by \(S^7 \subset \mathbb{H}^2 \to S^4 = \mathbb{H}\cup\infty\)
    • \(S^7 \to S^{15} \to S^8\)
      • Induced by \(S^{15} \subset\mathbb{O}^2 \to S^8 = \mathbb{O}\cup\infty\)

  • \(SO^3 \to SO^4 \to S^3\)

  • \(U^{n-1} \to U^n \to S^{2n-1}\)

    • Can compute \(H^*(U^n)\)

  • \(\Omega S^n \to PS^n \to S^n\), path-loop fibration

    • \(\Omega S^3 \to PS^3 \to S^3\):
      • Can compute \(H^*(\Omega S^n)\)

  • \(Y \to X\times Y \to X\) (not used here)

  • Fibrations

  • \(SO_{n-1}(R) \to SO_n(R) \to S^{n-1}\)

  • \(S^n \xrightarrow{E} \Omega S^{n+1} \xrightarrow{H} \Omega S^{2n+1}\)

  • \(S^1 \to S^{2n+1} \to {\mathbb{CP}}^n\)

  • \(\Omega B \to PB \to B\)

  • \(K(A, n) \to K(B, n) \to K(C,n)\) for any SES of groups.

  • \(S^0 \to S^1 \to {\mathbb{RP}}^1 = S^1\)

  • \(S^1 \to S^3 \to {\mathbb{CP}}^1 = S^2\)

  • \(S^3 \to S^7 \to \mathbb{HP}^1 = S^4\)

  • \(S^7 \to S^{15} \to \mathbb{OP}^1 = S^8\)

Define the Stiefel Manifold: \begin{align*} \mathbb{V}(k, n) = \left\{{A \in \mathbb{F}^{nk}\mathrel{\Big|}A \mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu^t = I}\right\} \end{align*}

and the Grassmanian

\begin{align*} G(k, n) = ? \end{align*}

Obtained from fiber bundles involving Stiefel Manifold :

  • \(O^{n-1} \to O^n \to S^{n-1}\)
  • \(SO^{n-1} \to SO^n \to S^{n-1}\)
  • \(U^{n-1} \to U^n \to S^{2n-1}\)
  • \(SU^{n-1} \to SU^n \to S^{2n-1}\)
  • \(Sp^{n-1} \to Sp^n \to S^{4n-1}\)
  • \(SO^n \to O^n \to S^0\)
  • \(SU^n \to U^n \to S^1\)
  • \(\mathbb{V}(k, k) \to \mathbb{V}(k, n) \to \mathbb{G}(k, n)\)

Interesting Spaces to Look At:

\(O, SO, Spin, U, or Sp\)

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