Intuitive

Concrete

Definition Two maps $f$ and $g$ are called homotopic, denoted by $f \tilde g$ if we can continuously transform them into each other. (See the example below).

Source: page 14 in https://arxiv.org/pdf/hep-th/0403286.pdf

Example

Equivalence of $f(x)=x$ and $g(x)=$const.

The two maps

$$ \mathbb{R}^n \to \mathbb{R}^n : f(x)= x $$ and $$ \mathbb{R}^n \to \mathbb{R}^n : g(x)= x_0 = \text{const} $$

are homotopic, because we can define

$$ F(x,t) = (1-t)x+tx_0 ,$$

which continuously transforms $f(x)=F(x,0)$ into $g(x)=F(x,1)$.

Abstract

Why is it interesting?

The QCD vacuum can only be understood by classifying all $SU(3)$ gauge transformations in terms of homotopic equivalence classes.

In physics, we often can identify the parameter t as time. Classical fields, evolving continuously in time are examples of homotopies. Here the restriction to continuous functions follows from energy considerations. Discontinuous changes of fields are in general connected with infinite energies or energy densities. […] Homotopy theory classifies the different sectors (equivalence classes) of field configurations. Fields of a given sector can evolve into each other as a function of time. https://arxiv.org/pdf/hep-th/0403286.pdf

The language of homotopy groups is not just to impress people, but gives us a unifying langue to discuss topological solitons.

page 283 in QFT in a Nutshell by A. Zee

In topology and in physics, an important equivalence relation is that of homotopy. In quantum field theory, topological defects arise as solutions to partial differential equations that are homotopically distinct from the vacuum solution. This can happen when the manifold specifying the boundary conditions of space-time has a nontrivial homotopy group. The equations preserve topological structure, so solutions can be classified by their homotopy class. Because two homotopically distinct manifolds are not homeomorphic, i.e., they cannot be continuously deformed into one another, two homotopically distinct solutions to the field equations can not be transformed into one another without a large energy input (which we will calculate in a simple example).

http://www.dartmouth.edu/~dbr/topdefects.pdf