EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 10, December 22th 2005

Solutions due on Thursday, 12th of January.

Exercise 10.1. Show that the dual of a Hopf-algebra of finite type over a field F is a Hopf-algebra

Exercise 10.2. Show that ℝP^{∞} and ℂP^{∞} are associative and commutative Hspaces. Hint: represent an element of 𝕂P^{∞} by a non-zero polynomial and use multiplication of polynomials.

Exercise 10.3. Compute the Pontrjagin product on the groups H_{∗}(ℝP^{∞}; 𝔽_{2}) and for H_{∗}(ℂP^{∞}; ℤ).

Exercise 10.4. Compute the Pontrjagin product of H_{∗}(T^{n}; ℤ), where T^{n} is the topological group S^{1} × ⋯ × S^{1} (n-factors).

Joyeux Noël et bonne année 2006 !

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