91Ó°ÊÓ

Let \(M\) be a closed orientable surface embedded in \(\mathbb{R}^{3}\) in such a way that reflection across a plane \(P\) defines a homeomorphism \(r: M \rightarrow M\) fixing \(M \cap P,\) a collection of circles. Is it possible to homotope \(r\) to have no fixed points?

Show that any two reflections of \(S^{n}\) across different \(n\) -dimensional hyperplanes are homotopic, in fact homotopic through reflections. [The linear algebra formula for a reflection in terms of inner products may be helpful.]

A polynomial \(f(z)\) with complex coefficients, viewed as a map \(\mathrm{C} \rightarrow \mathrm{C}\), can always be extended to a continuous map of one- point compactifications \(\hat{f}: S^{2} \rightarrow S^{2}\). Show that the degree of \(\hat{f}\) equals the degree of \(f\) as a polynomial. Show also that the local degree of \(\hat{f}\) at a root of \(f\) is the multiplicity of the root.

Compute the homology groups of the following 2-complexes: (a) The quotient of \(S^{2}\) obtained by identifying north and south poles to a point. (b) \(S^{1} \times\left(S^{1} \vee S^{1}\right)\) (c) The space obtained from \(D^{2}\) by first deleting the interiors of two disjoint subdisks in the interior of \(D^{2}\) and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise oricntations of these circles. (d) The quotient space of \(S^{1} \times S^{1}\) obtained by identifying points in the circle \(S^{1} \times\left\\{x_{0}\right\\}\) that differ by \(2 \pi / m\) rotation and identifying points in the circle \(\left\\{x_{0}\right\\} \times S^{1}\) that differ by \(2 \pi / n\) rotation.

Show the isomorphism between cellular and singular homology is natural in the following sense: \(A\) map \(f: X \rightarrow Y\) that is cellular \(-\) satisfying \(f\left(X^{n}\right) \subset Y^{n}\) for all \(n-\) induces a chain map \(f_{*}\) between the cellular chain complexes of \(X\) and \(Y,\) and the map \(f_{*}: H_{n}^{C W}(X) \rightarrow H_{n}^{C W}(Y)\) induced by this chain map corresponds to \(f_{*}: H_{n}(X) \rightarrow H_{n}(Y)\) under the isomorphism \(H_{n}^{C W} \approx H_{n}\).

For \(X\) a finite \(\mathrm{CW}\) complex and \(p: \tilde{X} \rightarrow X\) an \(n\) -sheeted covering space, show that \(\chi(\tilde{X})=n \chi(X)\).

Show that the second barycentric subdivision of a \(\Delta\) -complex is a simplicial complex. Namely, show that the first barycentric subdivision produces a \(\Delta\) -complex with the property that each simplex has all its vertices distinct, then show that for a \Delta-complex with this property, barycentric subdivision produces a simplicial complex.

Show that \(S^{1} \times S^{1}\) and \(S^{1} \vee S^{1} \vee S^{2}\) have isomorphic homology groups in all dimensions, but their universal covering spaces do not.

Use the Mayer-Vietoris sequence to show there are isomorphisms \(\tilde{H}_{n}(X \vee Y) \approx\) \(\tilde{H}_{n}(X) \oplus \tilde{H}_{n}(Y)\) if the basepoints of \(X\) and \(Y\) that are identified in \(X \vee Y\) are defor mation retracts of neighborhoods \(U \subset X\) and \(V \subset Y\).

Give an elementary derivation for the Mayer-Vietoris sequence in simplicial homology for a \(\Delta\) -complex \(X\) decomposed as the union of subcomplexes \(A\) and \(B\).