Homotopy

Topological spaces are enormously varied and homeomorphisms in general give much too fine a classification to be useful. Algebraic topology involves the classification of topological spaces in terms of algebraic objects (groups, rings) that are invariant under usefully large classes of homeomorphisms. The fundamental concept here is that of homotopy equivalence, for maps and spaces. Homotopy is studied in detail in Dodson and Parker [4], which contains many applications. We follow that approach here. A pair of continuous maps $f,g : X \rightarrow Y$ which agree on $A \subseteq X$ is said to admit a homotopy H from f to g relative to A if there is a map

$$X\times [0,1] \stackrel{H}{\longrightarrow} Y : (x,t) \longmapsto H_{t}(x)$$

with H_t(a)=H(a,t) = f(a)=g(a) for all $a\in A$, $H_0 = H(\ ,0) = f$, and $H_1 = H(\ ,1) = g$. Then we write $f\stackrel{H}\sim g \ (rel A)$. If $A = \emptyset$ or A is clear from the context (such as $A={\ast}$ for pointed spaces when A is a point), then we write $f\stackrel{H}\sim g$, or sometimes just $f\sim g$ and say that f and g are homotopic. We can also think of H as either of:

• a 1-parameter family of maps
  
  $$\{H_{t} : X \longrightarrow Y \ \mid \ t \in [0,1]\ \}\ \ \mbox{with} \ \ H _{0} = f \ \ \mbox{and} \ \ H _{1} = g\, ;$$
  
  

• a curve c_H from f to g in the function space Y^X of maps from X to Y
  
  $$c_{H} : [0,1] \longrightarrow Y^X : t \longmapsto H_{t}\,.$$
  
  

We call f nullhomotopic or inessential if it is homotopic to a constant map. Intuitively, we picture H as a continuous deformation of the graph of f into that of g. Exercises

1.

Use the standard homeomorphism

$$h : [a,b]\longrightarrow [0,1] : s\longmapsto \frac{s-a}{b-a}$$

to show that

$$f : [0,1]\longrightarrow [0,1] : s\longmapsto \left\{ \begi... ...pt] (s+1)/2 \ & s\in [\frac{1}{2},1] \end{array} \right.$$

is homotopic to the identity on [0,1]. Deduce that being homotopic is a transitive relation on paths and on loops in any space. Observe that a loop in X is a path

$$c: [0,1] \longrightarrow X \ \ \ \mbox{with}\ c(0)=c(1)\,,$$

so for loops we are interested in homotopy $rel\{0,1\}$.

2.

Supply the proof that $\sim$ determines an equivalence relation.

3.

Consider the identity map, $1_{\mathbb{S} ^1}:\mathbb{S} ^1\rightarrow \mathbb{S} ^1$, as a closed curve on the torus $\mathbb{S} ^1\times~\mathbb{S} ^1$ and find two other closed curves on the torus such that all three belong to different homotopy classes.

4.

If two continuous maps $f,g : X\rightarrow \mathbb{S} ^{n}$ have $f(x) \neq -g(x)$ for all $x \in X$ then f and g are homotopic. For, otherwise consider

$$\frac{tf + (1-t)g}{\Vert tf + (1-t)g\Vert}\,.$$

5.

Any two continuous maps into a contractible space are homotopic.

Two topological spaces X, Y are said to be of the same homotopy type or homotopy equivalent or, loosely, just homotopic if there exist (continuous) maps

$$f:X \longrightarrow Y, \ \ g:Y \longrightarrow X$$

$$\mbox{with} \ \ gf \sim 1_X \ \ \mbox{and} \ \ fg \sim 1_Y\,.$$

Then we write $X \simeq Y$ and say that f and g are mutual homotopy inverses or inverse up to homotopy. Similarly to the case for maps, $\simeq$ is an equivalence relation on any collection of topological spaces and one sometimes speaks (loosely) of spaces in the same class as being homotopic. The spaces in the homotopy equivalence class determined by a singleton space are called contractible; we often use $\ast$ to denote a singleton space. It turns out that equivalences up to homotopy are sufficient for easy proof of a wide range of important results in topology and analysis, (like fixed point theorems, extension and lifting theorems, fundamental theorem of algebra, hairy ball theorem ...) as may be seen in [4]. Exercises
The following X,Y are homotopically equivalent spaces which are not homeomorphic in the usual topologies (cf. Figure 1). Here, the wedge product $\mathbb{S} ^1\vee \mathbb{S} ^1$ is the quotient of the disjoint union of two circles, obtained by identifying one point in each circle to each other.

  

Figure 1: Homotopy equivalent spaces that are not homeomorphic

1.

$X = \mathbb{S} ^{n}$, $Y = \mathbb{S} ^{n}\times \mathbb{R} ^{m}$;

2.

$X = \mathbb{R} ^{n}$, $Y = \{0\}$;

3.

$X = \mathbb{S} ^{n-1}$, $Y = \mathbb{R} ^{n}\setminus {0}$;

4.

$X = \mathbb{S} ^{1}\vee \mathbb{S} ^{1}$, Y = punctured Klein bottle;

5.

$X = \mathbb{S} ^{1}\vee \mathbb{S} ^{1}$, Y = punctured torus.