6520-2022-08

Let \(X\) be the compact surface obtained from the disk by deleting two disjoint open disks, as in the figure below. Fix orientations on two boundary components as in the figure. Identify the boundaries using each of the two possible orientations on the third component. Show that the two quotient spaces so obtained have non-isomorphic fundamental groups. Hint: abelianize.

Figure: a disk with two disjoint open disks removed; the two inner boundary components have the indicated orientations.

Draw a picture of a covering space of \(S^1\vee S^1\) whose fundamental group is the normal subgroup of \(\pi _1(S^1\vee S^1,*)\) generated by \(a^2\), \(b^2\), and \((ab)^2\), and prove your covering space is the correct one.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). Explicitly describe the attaching maps of all cells. Use this cell structure to compute the homology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}/2\).

Compute the homology groups of the cell complex obtained by identifying the sides of the polygon below as indicated:

Figure: a hexagon with boundary edges, in clockwise order starting at the top, labeled \(a^{-1}\), \(c\), \(b\), \(a\), \(c^{-1}\), and \(b^{-1}\), with arrows indicating the side identifications.

Let \(M\) be closed, connected, 5-manifold such that \(\pi _1(M)\cong \mathbb {Z}/7\) and \(H_2(M)=\mathbb {Z}\). Compute all homology and cohomology groups of \(M\) with \(\mathbb {Z}\)-coefficients.

Let \(S^n\) be the \(n\)-sphere, for some integer \(n\ge 1\), and \(f:S^n\to S^n\) a continuous map.

1. Show that if \(f\) is not surjective, then it is homotopic to a constant map.
2. Construct an example of a surjective map \(f\) that is homotopic to a constant map.
3. Is every map \(f\) as above homotopic to a constant? Either give a proof or a counterexample.