6520-2020-08

Let \(X\) be a connected cell complex and \(f:\mathbb {R}P^{2n}\to X\) a covering map from the real projective space of dimension \(2n\). Show that \(f\) is a homeo- morphism.

Let \(x_0\in \mathbb {R}P^2\) be a basepoint and view the wedge \(\mathbb {R}P^2\vee \mathbb {R}P^2\) as the subspace of the product \(\mathbb {R}P^2\times \mathbb {R}P^2\) where at least one coordinate is \(x_0\).

1. Use the Seifert-van Kampen theorem to compute \(\pi _1(\mathbb {R}P^2\vee \mathbb {R}P^2)\).
2. Prove that \(\mathbb {R}P^2\vee \mathbb {R}P^2\) is not a retract of \(\mathbb {R}P^2\times \mathbb {R}P^2\).

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

Recall that the mapping torus of a map \(f:X\to X\) is the quotient space \(T_f\) obtained from \(X\times [0,1]\) by identifying \((x,1)\) with \((f(x),0)\) for every \(x\in X\). Compute the homology groups of \(T_f\) when \(f:S^n\to S^n\) is a map of degree \(d\) and \(n > 1\).

Let \(G\) be a finite group of order \(d\) and \(\phi :F_n\to G\) an epimorphism from the free group of rank \(n\). Show that \(\operatorname {Ker}(\phi )\) is a free group and compute its rank.

Let \(S_g\) denote an orientable surface of genus \(g\). So for example \(S_0\) is the 2-sphere and \(S_{1}\) is the torus.

1. If \(g>0\) show that every map \(S_0\to S_g\) is null-homotopic.
2. If \(g>h>0\) show that every map \(S_h\to S_g\) has degree 0, but show by example that there exist maps that are not null-homotopic.