EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 3, November 3th 2005

Solutions due on Thursday, 10th of November.

Exercise 3.1. Let ℓ, m, n ⩾ 2 be integers. Compute the following Ext-groups.

(a) Ext_{ℤ}^{∗}(ℤ/m; ℤ),

(b) Ext_{ℤ}^{∗}(ℤ/m; ℤ/n),

(c) Ext_{ℤ/ℓ}^{∗}(ℤ/m, ℤ/n) when defined,

(d) Ext_{ℤ}^{∗}(ℚ, ℤ). For (d), it might be useful to consider the short exact sequence of ℤ-modules

0 → ℤ → ℚ → ℚ/ℤ → 0

and the decomposition of ℚ/ℤ as a sum over the prime numbers p of the Prüfer groups ℤ/p^{∞}.

Exercise 3.2. Repeat the exercise above, replacing Ext_{R}^{∗} by Tor_{∗}^{R}.

Exercise 3.3. Explain how the cohomology mod p Bockstein homomorphisms interact with the cup product.

Typeset by AMS-TEX