Tags: #projects/notes/reading #projects/active #projects/notes Refs: spectral sequence

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
• \(S^n \subset {\mathbf{R}}^{n+1}\) and \(S^{2n-1} \subset {\mathbf{C}}^n\)
• 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
  • \((rm)\otimes n = r(m\otimes n) =m \otimes(rn)\)
  • \((r+s)(m\otimes n) = rm\otimes n + sm\otimes n\)
  • \({\mathbf{Z}}_p \otimes_{\mathbf{Z}}{\mathbf{Z}}_q = {\mathbf{Z}}/\gcd(p,q)\) and \(\gcd(p,q) = 1\) yields 0.
  • Some computations:
    • \({\mathbf{Z}}_n \otimes_{\mathbf{Z}}{\mathbf{Q}}= 0\)
    • \({\mathbf{Z}}_n \otimes_{\mathbf{Z}}{\mathbf{Q}}/{\mathbf{Z}}= 0\)
    • \({\mathbf{Q}}\otimes_{\mathbf{Z}}{\mathbf{Q}}= {\mathbf{Q}}\)
    • \(({\mathbf{Q}}/{\mathbf{Z}})\otimes_{\mathbf{Z}}{\mathbf{Q}}= 0\)
    • \({\mathbf{Q}}/{\mathbf{Z}}\otimes_{\mathbf{Z}}{\mathbf{Q}}/{\mathbf{Z}}= 0\)
    • \(R[x]\otimes_R S \cong S[x]\)
    • \(k \to K\) a field extension: \(k[x]/(f) \otimes_k K \cong K[x]/(f)\)
  • Symmetric, Associative
  • \((\oplus A_i )\otimes B = \oplus(A_i \otimes B)\)
  • \({\mathbf{Z}}\otimes A = A\)
  • \({\mathbf{Z}}_n \otimes A = \frac{A}{nA}\)

List of Results

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

List of Fibrations

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

      {\mathbf{R}}

      ^2

      \to 

      S^1 =

      {\mathbf{R}} 

      \cup 

      \infty 

      $
  • \(S^1 \to S^3 \to S^2\)
    • Induced by \(S^3 \subset {\mathbf{C}}^2 \to S^2 = {\mathbf{C}}\cup\infty\)
  • \(S^3 \to S^7 \to S^4\)
    • 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

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

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

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

• \(\Omega B \to PB \to B\)

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

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

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

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

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

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:

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