Poincaré duality (Ex)
From Manifold Atlas
Let the local coefficient system of twisted the orientation character of a compact manifold , let denote the total space of the non-trivial linear sphere bundle over and let be a closed simply connected manifold.
Determine the following homology groups (in terms of where appropriate):
Determine the following cohomology groups and verify Poincaré duality using the homology computations above:
Here are some helpful references for the definitions involved: [Davis&Kirk2001, Ch 5] [Wall1967a, Chapter 1]