6520-2021-05

Consider the group \(G\) given by the presentation

where \(p\), \(q\) are given integers.

1. Construct a (connected) CW-complex \(X\) with a basepoint \(x\) such that \(\pi _1(X,x)\cong G\).
2. Compute \(H_1(X)\) and give conditions on \(p\), \(q\) that are equivalent to \(H_1(X)\) being torsion-free.

Let \(p : \tilde {X} \to X\) be a covering map between two connected CW-complexes. If \(p\) is null-homotopic, prove that \(\tilde {X}\) is contractible. If you use a theorem make sure you state it carefully and include all the hypotheses.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). For each cell define its attaching map.

Suppose a space \(Y\) is obtained from a space \(X\) by attaching a single \(n\)-cell

via an attaching map \(f:S^{n-1}\to X\), i.e.

Show that \(H_i(X)\cong H_i(Y)\) for all \(i\) except at most two and give an example where there are two values of \(i\) where the two groups are not isomorphic.

Let \(M\) be a closed connected 5-manifold such that \(H_1(M)\cong \mathbb {Z}/3\) and \(H_2(M)\cong \mathbb {Z}\). Compute \(H_i(M)\) for all \(i\).

1. Define the notion of a chain morphism between two chain complexes and prove that it induces a homomorphism in homology.
2. Define the notion of a chain homotopy equivalence.
3. Prove that a chain homotopy equivalence induces an isomorphism in homology.