Poincaré duality (Ex)

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Let \Zz_\omega the local coefficient system of \Zz twisted the orientation character \omega \colon \pi_1(M) \to \Zz/2 of a compact manifold M, let S^{n-1} \tilde \times S^1 denote the total space of the non-trivial linear sphere bundle over S^1 and let N be a closed simply connected manifold.

Determine the following homology groups (in terms of H_*(N; \Zz where appropriate):

  1. H_*(\RP^n; \Zz)
  2. H_*(\RP^n; \Zz_\omega)
  3. H_*(S^{n-1} \tilde \times S^1; \Zz)
  4. H_*(S^{n-1} \tilde \times S^1; \Zz_\omega)
  5. H_*(\RP^n \sharp N; \Zz_\omega)
  6. H_*((S^{n-1} \tilde \times S^1) \sharp N; \Zz_\omega)
  7. H_*(S^1 \times N; \Zz_\omega)
  8. H_*(\RP^n \times N; \Zz_\omega)

Determine the following cohomology groups and verify Poincaré duality using the homology computations above:

  1. H^*(\RP^n; \Zz_\omega)
  2. H^*(\RP^n; \Zz)
  3. H^*(S^{n-1} \tilde \times S^1; \Zz_\omega)
  4. H^*(S^{n-1} \tilde \times S^1; \Zz)
  5. H^*(\RP^n \sharp N; \Zz)
  6. H^*((S^{n-1} \tilde \times S^1) \sharp N; \Zz)
  7. H^*(S^1 \times N; \Zz)
  8. H^*(\RP^n \times N; \Zz)

Here are some helpful references for the definitions involved: [Davis&Kirk2001, Ch 5] [Wall1967a, Chapter 1]

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