Exercises

1. We have obtained $S^2$ by attaching $E^2$ to a singleton with the attaching map as the constant map on the boundary of $E^2$. Discuss how would you obtain $S^n$ analogously as an adjunction space.
2. Show that if $X$ and $B$ are connected/path-connected then $X\sqcup_f B$ is connected/path-connected.
3. Describe the push out resulting from the diagram
   $\begin{CD} S^{n-1} @> i_1 >> E^n \\ @V{i_2}VV @.\\ E^n @. \\ \end{CD}$
4. Show that $S^m\times S^n$ results from attaching an $n+m$ cell to $S^n\vee S^m$. Hint: Let $I$ denote $[0, 1]$ and define a map $f: \partial(I^n\times I^m) \longrightarrow S^n\vee S^m$ as follows
   $$f(z) = \left\{\begin{array}{lll} (\eta_1(x), y_0) & & \mbox{if }... ...\\ (x_0, \eta_2(y)) & & \mbox{if }\;y\in\partial I^m \\ \end{array} \right.$$
   and $\eta_1:I^n\longrightarrow S^n$ and $\eta_2:I^m\longrightarrow S^m$ are the quotient maps of exercise 1.
5. Prove theorem (25.3).
6. Fill in the details in examples (25.4) and (25.5).

in

Lecture - XXVI Seifert Van Kampen theorem and knots

in This is one of the most famous theorem concerning the fundamental group which serves as a tool for computations and applications to combinatorial group theory. If $U$ and $V$ are path connected open subsets of a topological space such that $U\cap V$ is path connected, the theorem provides information on the geometry of $U\cup V$ in terms of the geometry of $U$, $V$ and $U\cap V$. In precise terms it states that the $\pi_1$ functor maps the push-out diagram of pointed topological spaces with $x_0\in U\cap V$,

$\begin{CD} (U\cap V, x_0) @> i_1 >> (U, x_0) \\ @V{i_2}VV @VV{j_1}V\\ (V, x_0) @> j_2 >> (U\cup V, x_0) \\ \end{CD}$

to the push-out diagram of groups:

$\begin{CD} \pi_1(U\cap V, x_0) @> {i_1}_* >> \pi_1(U, x_0) \\ @V{{i_2}_*}VV @VV{{j_1}_*}V\\ \pi_1(V, x_0) @> {j_2}_* >> \pi_1(U\cup V, x_0) \\ \end{CD}$

thereby giving a precise description of the group $\pi_1(U\cup V, x_0)$ in terms of the groups $\pi_1(U, x_0)$, $\pi_1(V, x_0)$ and $\pi_1(U\cap V, x_0)$. Thus $\pi_1(U\cup V, x_0)$ is the free product of $\pi_1(U, x_0)$ and $\pi_1(V, x_0)$ amalgamated along $\pi_1(U\cap V, x_0)$. The theorem enables us to calculate quickly the fundamental groups of several important spaces.