Homotopy theory, which is the main part of algebraic topology, studies topological objects up to homotopy equivalence.

## Introduction to Homotopy Theory

Homotopy equivalence is weaker relations than topological equivalence, i. Even though the ultimate goal of topology is to classify various classes of topological spaces up to a homeomorphism, in algebraic topology, homotopy equivalence plays a more important role than homeomorphism, essentially because the basic tools of algebraic topology homology and homotopy groups are invariant with respect to homotopy equivalence, and do not distinguish topologically nonequivalent, but homotopic objects.

The idea of homotopy can be turned into a formal category of category theory. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.

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Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed. It is not an easy book and definitely depends on the level of the student but there are undergrad students who could do it and benefit a lot from it with hard work. Pece Pece 8, 1 1 gold badge 13 13 silver badges 41 41 bronze badges.

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## Course on Homotopy Theory (first semester 2012/2013)

Summary Notes for a second-year graduate course in advanced topology at MIT, designed to introduce the student to some of the important concepts of homotopy theory. Share Share Share email. Authors George W. Aumann Cart Buying Options.