Homework number 4
Do one of the exercises 10 and 11 and the exercise 12.
Exercise 10. Compute Hn(RP10
/RP5
) using the CW-structure of this space.
Exercise 11. Let X = Dn+1
∪f Sn
, where f : Sn
→ Sn
is a map of degree k. Compute
Hi(X) and determine all nontrivial homomorphisms in the long exact sequence for the
couple (X, Sn
).
Exercise 12. Let x be a point in a space X. The local homology groups of X in the
point x are the groups Hn(X, X − {x}). Using excission theorem they are isomorphic
to Hn(U, U − {x} where U is an open neighbourhood of the point x. These local
homology groups are preserved by homeomorphisms and so they are important if you
want to decide if two spaces are homeomorphic.
(a) Consider X as a one-skeleton of tetrahetron A0A1A2A3 and compute the local
homology groups of this space with respect to a vertex of the tetrahedron and with
respect to a center of an edge.
(b) Now add the center T of the tetrahedron and let Y be the union of all six triangles
TAiAj, i = j. Compute the local homology groups of Y with respect to T.
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