Introduction to Homotopy Theory

Abstract

Consider two manifolds X and Y together with a set of continuous maps f, g, …

$$\displaystyle{ f:\,\, X \rightarrow Y \;,\quad x \rightarrow f(x) = y\;;\quad x \in X\;,\quad y \in Y \;. }$$

Then two maps are defined to be homotopic if they can be continuously distorted into one another. That is, f is homotopic to g, $f \sim g,$ if there exists an intermediate family of continuous maps $H(x,t),\,0\leqslant t\leqslant 1,$

$$\displaystyle{ H:\,\, X \times I \rightarrow Y \;,\quad I = [0,1] }$$

(27.1)

such that

$$\displaystyle{ H(x,0) = f(x)\;,\quad H(x,1) = g(x)\;. }$$

(27.2)

H is then called a homotopy between f and g.