Problem in Algebraic Topology

[220205]

Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you all!

Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f.
i). Calculate the integral homology of Xf .
ii). Find an n and an f such that H*(Xf ;Z=2) has a non-trivial product. Justify it.

Let X be a path-connected finite CW-complex and see S1 as the complex numbers
of norm 1. Let [X; S1] denote the set of homotopy classes [f] of maps f : X -> S1.
i). Show that the set [X; S1] has the structure of a group induced by the multiplication
in S1.
ii). Show that [X; S1] is naturally isomorphic to H1(X; Z) as groups.( use Eilenberg-Maclane space)

 

[pat_connell]

Let G be a discrete group and X be a G-space. Recall that the equivariant cohomology rings of X are defined as H*G(X;Z) = H*(X x_G EG; Z). i). Show that for any X, the natural map H*(X;Z) -> H*G(X;Z) is injective. ii). Give a necessary condition for H*G(X;Z) to be a free abelian group. iii). For what groups G is H*G(X;Z) free?