Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
Two continuous functions are called homotopic if one of them can be continuously deformed into the other. A homotopy equivalence is then a map $f:X \rightarrow Y$ admitting a "homotopy inverse", i.e. a map $g:Y \rightarrow X$ such that $gf$ is homotopic to $\mbox{id}_X$ and $fg$ is homotopic to $\mbox{id}_Y$. Broadly speaking, then, homotopy theory is the study of topological spaces up to homotopy equivalence. As always, when one chooses to ignore certain aspects of the objects under study, other properties come to the fore. The first example of a homotopy-invariant property is that of the fundamental group.
