Introduction to paths H and application of homotopy theory in physics

Firstly, we introduced the action of space operators on a regular interval to generate a variable interval. Secondly, we introduced the concept of a family T of paths H, and we showed that these paths are homotopic on a contractible space even though they do not have common endpoints. Finally, we applied the concept of paths H on a contractible space in physics. Let A be a subset of X. Let I$_{\mathrm{\{a,b\}\thinspace }}$be a regular interval such that \textbraceleft I$_{\mathrm{\{a,b\}}}$\textbraceright $\subseteq $ A, for a, b $\in $ A. Let ($\alpha_{\mathrm{a}}$,$_{\mathrm{\beta \thinspace b}})$ be space operators associated with\textbraceleft a,b\textbraceright , then a variable interval is I$_{\mathrm{\{x,y\}}}=(\alpha_{\mathrm{a}}$,$\beta _{\mathrm{\thinspace b}})$I$_{\mathrm{\{a\thinspace ,b\}}}$ such that \textbraceleft I$_{\mathrm{\{x,y\}}}$\textbraceright $\subseteq $ X, min\textbraceleft I$_{\mathrm{\{x,y\}}}$\textbraceright $=$ax, and max\textbraceleft I$_{\mathrm{\{x,y\}}}$\textbraceright $=$by for all x, y $\in $X. Let X be a topological space. Let f, g: [0,1] $\to $X be continuous paths for all t $\in $ [0,1]. T is the family of continuous paths H: [0,1]x[0,1]$\to $ X such that H(t,0)$=$f, H(t,1)$=$g for all t $\in $ [0,1], and H(0,s$_{\mathrm{f}})=$f(0), H(1,s$_{\mathrm{f}})=$f(1), H(0, s$_{\mathrm{g}})=$g(0), H(1,s$_{\mathrm{g}})=$g(1) for all s$_{\mathrm{f,\thinspace }}$s$_{\mathrm{g\thinspace \thinspace }}\in $ [0,1]. Such f and g are H-topic paths. If X is contractible, then H is a homotopy. In addition, if s$_{\mathrm{f}}=$s$_{\mathrm{g}}$, then f(0)$=$g(0) and f(1)$=$g(1), and the family T of paths H becomes the well-known homotopy of paths (with same endpoints). Let M$_{\mathrm{G}}$ be a simply connected gravitational field. We showed that the Hamiltonian for free fall-paths on M$_{\mathrm{G}}$ obeys the homotopy theory.