Modification of the notion of homotopy for paths. vid. We don't allow to move the end points

Definition

Let $f,g: I \to X$ be paths with the same initial and final points $f(0) = g(0)$ and $(1) = g(1)$. Then $f$ and $g$ are (path-)homotopic if there is a continuous map $F: I \times I \to X$ s.t.

• $F(s,0) = f(s)$
• $F(s,1) = g(s)$
• $F(0,t) = f(0)=g(0)$
• $F(1,t) = f(1) = g(1)$

This is an Equivalence relation on the set of paths in $X$from a fixed initial point to a fixed end point.

Define Product of paths

Def For a path $f: I \to X$, let $[f]$ be the class equivalence class of $f$ w.r.t. path homotopy.

Let $f,g$ be pathes with $f(1)=g(0)$, define $[f] * [g] = [f*g]$ (product of equivalence classes of paths, which is well defined).

To define a group we need to show.

Associativity

Unit element

Inverse