The most immediate way to use classical homotopy theory to study topological spaces endowed with extra structures, such as complex spaces, is to forget these structures and study the underlying topological space. Usually this procedure is inadequate since what we are forgetting is not homotopy invariant nor topological invariant. There are more sophisticated approaches to use homotopy theory to try to detect different complex structures on the same topological space which prove to be effective, for instance if the topological space is a complex variety and the complex structures endow the tangent bundle of different Chern classes/numbers. A much more thorough approach can be achieved by using model category theory to provide the category of complex spaces with holomorphic maps of an homotopy theory which realizes to the ordinary topological one, it is biholomorphic but not topological invariant. These techniques have been previously implemented by Morel and Voevodsky to create the so called A^1 homotopy theory of algebraic varieties few years ago.

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Last updated: 02 Nov 2024