NoteNextra-origin/content/Math4202/Math4202_L26.md

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

f_*=\pi_1(X,f(x_0))\quad\text{and}\quad g_*=\pi_1(Y,g(x_0))

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.

Proof

$I\times I\xrightarrow{\gamma_{id}} X\times I\xrightarrow{H} Y$

• I\times \{0\}\mapsto f\circ\gamma
• I\times \{1\}\mapsto g\circ\gamma
• \{0\}\times I\mapsto \alpha
• \{1\}\times I\mapsto \alpha

As I\times I is convex, I\times \{0\}\simeq (\{0\}\times I)*(I\times \{1\})*(\{1\}\times I).

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.

Proof

\hat{\alpha} is an isomorphism of \pi_1(Y,y_0) to \pi_1(Y,y_1).

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

f_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)

is an isomorphism.

Proof

Let g:Y\to X be the homotopy inverse of f.

Then,

f_*\circ g_*=\alpha \circ id_{\pi_1(Y,y_0)}=\alpha

And g_*\circ f_*=\bar{\alpha}\circ id_{\pi_1(X,x_0)}=\bar{\alpha}

So f_*\circ (g_*\circ \hat{\alpha}^-1)=id_{\pi_1(X,x_0)}

And g_*\circ (f_*\circ \hat{\alpha}^-1)=id_{\pi_1(Y,y_0)}

So f_* is an isomorphism (have left and right inverse).

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

i_*:\pi_1(U,x_0)\to \pi_1(X,x_0)\quad j_*:\pi_1(V,x_0)\to \pi_1(X,x_0)

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