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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)