6520-2022-05

Let \(W\) be the union of the three coordinate axes in \(\mathbb {R}^3\) and set \(X:=\mathbb {R}^2\setminus W\); thus \(X\) consists of points \((x,y,z)\) where none of the coordinates are zero. Compute \(\pi _1(X)\).

Let \(f:M\to N\) be a covering map between manifolds that are both nonempty, connected, and compact without boundary. Suppose \(f\) is homotopic to a constant map. Prove that \(N\) is a point.

Let \(X\) be a CW complex consisting of one vertex p, two edges a and b attached to \(p\) along their boundaries, and two faces \(f_1\) and \(f_2\), attached along \(ab^2\) and \(ba^2\), respectively.

1. Find \(\pi _1(X)\). Is it a finite group?
2. Compute the homology groups \(H_i(X;\mathbb {Z})\) of \(X\) for all \(i\).

Let \(T^2\) be the real 2-dimensional torus and \(f:S^1\vee S^1\to T^2\) a continuous map. Prove or disprove the following assertions:

1. There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(f\circ g:T^2\to T^2\) is homotopy to the identity map.

2. There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(g\circ f\) is homotopy to the identity map.

   Justify your answers.

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Let \(M\) be an orientable \(n\)-manifold that is connected, compact, and without boundary. Let \(f:S^n\to M\) be a continuous map such that the induced map \(f_*:H_n(S^n;\mathbb {Z})\to H_n(M;\mathbb {Z})\) is non-trivial. Calculate \(H^i(M;\mathbb {Q})\) and \(H_i(M;\mathbb {Q})\) for all \(i\).