Exercises

1. Prove that the map $\eta$ in the five lemma is surjective.
2. Show that the map (38.2) is indeed an isomorphism. To prove that it is surjective use the decompositions $S^{\;{\cal U}}(X) = S(X-U) + S($int$A)$ and $S^{\;{\cal V}}(A) = S(A-U) + S($int$A)$.
3. Prove the Barrett-Whitehead lemma.
4. Calculate the local homology groups $H_2(X, X-\{p\})$ in the following cases:

   (i)

   The space $X$ is the cylinder $S^1 \times [0, 1]$ and $p$ a point on its boundary.

   (ii)

   The space $X$ is the Möbius band and $p$ is a point on its boundary.

   Deduce that the cylinder and the Möbius band are not homeomorphic.
5. A topological manifold is a Hausdorff space in which each point has a neighborhood homeomorphic to an open ball in $\mathbb{R}^n$. Show that if $p$ is a point on a topological manifold $M$,
   $$H_n(M, \; M - \{p\}) \cong \mathbb{Z}.$$

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Lecture - XXXIX (Test - V)

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1. Calculate the homology groups of the double torus.
2. Show that any homeomorphism of $E^n$ onto itself must preserve the boundary.
3. Show that $\mathbb{R}P^n$ is not a retract of $\mathbb{R}P^{n+1}$. Use the lifting criterion.
4. Regard $S^2$ as the Riemann sphere and calculate the degree of the map $f:S^2\longrightarrow S^2$ given by $f(z)=z^n$.
5. Use the previous exercise to prove the fundamental theorem of algebra.
6. Show that $\mathbb{R}P^{2n}$ has the fixed point property. Does $\mathbb{R}P^3$ have the fixed point property?

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Lecture - XL Inductive limits

in We have frequently encountered situations where a certain space $X$ is canonically embedded in a larger space $Y$. A familiar example the sequence of orthogonal groups and the canonical inclusions

$$SO(2, \mathbb{R}) \longrightarrow SO(3, \mathbb{R}) \longrightarrow SO(4, \mathbb{R})\longrightarrow\dots \eqno(40.1)$$

where, the inclusion map $SO(n, \mathbb{R}) \longrightarrow SO(n+1, \mathbb{R})$ is given by

$$A \mapsto \begin{pmatrix} A & 0 \cr 0 & 1 \cr \end{pmatrix},\quad A \in SO(n, \mathbb{R})$$

The inductive limit of a sequence such (40.1) is a space which contains each individual member of the sequence, and is the smallest such space. The precise meaning of the adjective smallest would be clear from the formal definition that we shall presently give.

Let us look at a situation in the category of abelian groups. For a fixed prime $p$ let $C_{p^k}$ denote the cyclic group of order $p^k$. Then for each $j \leq k$, the group $C_{p^k}$ contains a (unique) cyclic group of order $p^j$ giving us a sequence of groups

$$C_p\longrightarrow C_{p^2}\longrightarrow C_{p^3} \longrightarrow \dots, \eqno(40.2)$$

in which the arrows inclusion maps. All these groups may be regarded as subgroups of $\mathbb{C} - \{0\}$ or as subgroups of the smaller group $S^1$. However there is a smallest group containing a copy of each the groups $C_{p^k}$ namely, the group

$$\Big\{ \exp\Big(\frac{2\pi i l}{p^k}\Big)\;/ \;l, k \in \mathbb{Z} \Big\} \eqno(40.3)$$

consisting of all $p^k$-th roots of unity ( $k = 1, 2, \dots$). This group (known as the Prüfer group) would then be the inductive limit of the family of cyclic groups $C_{p^k}$ ( $k = 1, 2, \dots$).

We now proceed to the formal definitions and prove the existence and uniqueness (upto isomorphism) of the inductive limit of a family of groups. We recall the notion of a directed set.