Math4202 Topology II (Lecture 26)

Algebraic Topology

Deformation Retracts and Homotopy Type

Lemma of homotopy equivalence

Let $f,g:X\to Y$ be continuous maps. let

And $H:X\times I\to Y$ is a homotopy from $f$ to $g$ with a path $H(x_0,t)=\alpha(t)$ for all $t\in I$.

Then $\hat{\alpha}\circ f_*=[\bar{\alpha}*(f\circ \gamma)*\alpha]=[g\circ \gamma]=g_*$. where $\gamma$ is a loop in $X$ based at $x_0$.

Corollary for homotopic continuous maps

Let $h,k$ be homotopic continuous maps. And let $h(x_0)=y_0,k(x_0)=y_1$. If $h_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)$ is injective, then $k_*:\pi_1(X,x_0)\to \pi_1(Y,y_1)$ is injective.

Corollary for nulhomotopic maps

Let $h:X\to Y$ be nulhomotopic. Then $h_*:\pi_1(X,x_0)\to \pi_1(Y,h(x_0))$ is a trivial group homomorphism (mapping to the constant map on $h(x_0)$).

Theorem for fundamental group isomorphism by homotopy equivalence

Let $f:X\to Y$ be a continuous map. Let $f(x_0)=y_0$. If $f$ is a homotopy equivalence  ($\exists g:Y\to X$ such that $fg\simeq id_X$, $gf\simeq id_Y$), then

is an isomorphism.

Fundamental group of higher dimensional sphere

$\pi_1(S^n,x_0)=\{e\}$ for $n\geq 2$.

We can decompose the sphere to the union of two hemisphere and compute $\pi_1(S^n_+,x_0)=\pi_1(S^n_-,x_0)=\{e\}$

But for $n\geq 2$, $S^n_+\cap S^n_-=S^{n-1}$, where $S^1_+\cap S^1_-$ is two disjoint points.

Theorem for “gluing” fundamental group

Suppose $X=U\cup V$, where $U$ and $V$ are open subsets of $X$. Suppose that $U\cap V$ is path connected, and $x\in U\cap V$. Let $i,j$ be the inclusion maps of $U$ and $V$ into $X$, the images of the induced homomorphisms

The image of the two map generate $\pi_1(X,x_0)$.