Exercises

1. Show that if $R^{\prime}$ and $R^{\prime\prime}$ are two reflections (each with respect to a coordinate plane) then they are conjugate by a homeomorphism. Deduce that both $R^{\prime}$ and $R^{\prime\prime}$ have degree $-1$.
2. Show that if a continuous map $f:S^n\longrightarrow S^n$ misses a point of $S^n$ then $f$ is homotopic to the constant map and so has degree zero.
3. Show that if $n$ is odd then the antipodal map of $S^n$ is homotopic to the identity map. Hint: Do it first for the case $n = 1$ and show that the homotopy may be achieved via a continuous rotation. The general case follows along similar lines by working with pairs of coordinates.
4. Show that $\mathbb{R}P^{2n}$ has the fixed point property.
5. Let $\eta:S^{2n}\longrightarrow \mathbb{R}P^{2n}$ be the covering projection. Show that $H_{2n}(\eta)$ is the zero map.
6. Show that the map (36.5) is a homeomorphism and (36.6) defines a continuous map. More generally given a continuous map $f : X \longrightarrow Y$ show that the composite
   $$\begin{CD} X\times [0, 1] @> {f\times \mbox{id}}>> Y\times [0, 1] @> >> \Sigma Y \end{CD}$$
   induces a map $\Sigma f:\Sigma X\longrightarrow \Sigma Y$. Imitate the computation in theorem [//] of lecture [//] to show that $H_{n+1}(\Sigma X) = H_n(X)$ when $n\geq 1$. What happens when $n = 0$?
7. Prove theorem (36.11). Note that the map $f : S^1 \longrightarrow S^1$ given by $f(z) = z^m$ has degree $m$.
8. Determine the degree of a polynomial as a map from $S^2$ to itself. Reprove the fundamental theorem of algebra.

in

Lecture - XXXVII Relative homology

in The homology groups $H_n(X)$ we have hitherto been studying are called the absolute homology groups. The relative homology groups $H_n(X, A)$ that we define here provide us a tool for understanding the geometry of a space $X$ in relation with its subspace $A$. This is facilitated by a long exact sequence in homology for the pair $(X, A)$. For instance if $A$ is a retract of $X$, this sequence breaks off into a bunch of short exact sequences each of which splits. The groups $H_n(X, A)$ are related to the absolute homology groups $H_n(X/A)$ for sufficiently well behaved pairs $(X, A)$ but we shall not get into this discussion here (see [16], p. 50).

Recall that if $A$ is a subspace of $X$ and $z$ is a non-trivial $n-$cycle in $A$ then it may be a boundary when viewed as a cycle in $X$. In other words, the inclusion map $i:A\longrightarrow X$ need not induce an injective map in homology. The relative homology group measures $H_{n+1}(X, A)$ the deviation from injectivity of the map $H_n(i)$.