## 10. Universal coefficient theorem

1. Identify the following abelian groups up to isomorphism.

(a). $\mathbb { Z } _ { m } \otimes \mathbb { Z } _ { n }$

(b). $\mathbb { Z } _ { 6 0 } ^ { 4 } \otimes \left( \mathbb { Z } _ { 2 4 } ^ { 3 } \oplus Z _ { 8 } ^ { 4 } \oplus \mathbb { Z } _ { 1 2 0 } \right)$

(c). $\mathbb { Z } _ { n } \otimes \mathbb { Q }$

(d). $\left( \mathbb { Z } \oplus \mathbb { Z } _ { n } \right) \otimes \left( \mathbb { Q } / \mathbb { Z } \right)$

2. (a). Compute Tor $( \mathbb { Z } \oplus \mathbb { Z } _ { 2 } \oplus \mathbb { Z } _ { 8 } , \mathbb { Z } \oplus \mathbb { Z } _ { 4 } \oplus \mathbb { Z } _ { 4 } )$

(b). Compute Ext(Z ⊕ $\mathbb { Z } _ { 2 } \oplus \mathbb { Z } _ { 3 } , \mathbb { Z } \oplus \mathbb { Z } _ { 4 } \oplus \mathbb { Z } _ { 5 } )$

3. Recall the calculation of the integer homology of $\mathbb { R } P ^ { n }$ by means of a cellular chain complex.
   Compute the following homology groups directly from the chain complex (and check them by means of the universal coefficient theorem).

(a). $H _ { * } ( \mathbb { R } P ^ { n } ; \mathbb { Z } _ { 2 } )$

(b). $H _ { * } ( \mathbb { R } P ^ { n } ; \mathbb { Z } _ { 3 } )$

(c). $H ^ { \ast } ( \mathbb { R } P ^ { n } ; \mathbb { Z } _ { 6 } )$

4. Show that the first cohomology group of any space is free abelian.

5. Show that for any space, the rational homology and cohomology can be computed via

$$
H _ { * } ( X ; \mathbb { Q } ) = H _ { * } ( X ; \mathbb { Z } ) \otimes \mathbb { Q } \qquad H ^ { * } ( X ; \mathbb { Z } ) = \operatorname { H o m } ( H _ { * } ( X ; \mathbb { Z } ) , \mathbb { Q } ) .
$$

6. Explain how to construct a space X having homology groups $H _ { i } ( X ; \mathbb { Z } )$ given by $\mathbb { Z } , \mathbb { Z } _ { 6 } , \mathbb { Z } _ { 1 2 } , \mathbb { Z } \oplus \mathbb { Z } _ { 4 }$ in dimensions $i = 0 \dots 3$ , and zero otherwise.
   Compute its cohomology groups $H ^ { * } ( X ; \mathbb { Z } )$

7. Compute the homology $H _ { * } ( \mathbb { R } P ^ { 2 } \times \mathbb { R } P ^ { 2 } ; \mathbb { Z } _ { 2 } )$

8. Compute the integral homology $H _ { * } ( \Sigma \mathbb { R } P ^ { 2 } \times \mathbb { R } P ^ { 2 } ; \mathbb { Z } )$ . (ΣX denotes the suspension of X, namely $X \times I$ with the ends $X \times \{ 0 \}$ and $X \times \{ 1 \}$ each collapsed to a (different) point).

9. Compute the integral homology $H _ { * } ( \mathbb { R } P ^ { 2 } \times \mathbb { R } P ^ { 3 } ; \mathbb { Z } )$

10. Show that if G is a topological group, then $H _ { * } ( G )$ has a natural (possibly non-commutative) algebra structure.
    Show also that there is a natural action of G on $H _ { * } ( G )$ , but that this action factors through the homomorphism $G \to \pi _ { 0 } ( G )$ , so that it’s trivial if G is path-connected.