Content: Algebraic topology is concerned with the construction of algebraic invariants usually groups associated to topological spaces which serve to distinguish between them. In essence, this means that they do not change under continuous deformation of the space and homotopy is a precise way of formulating the idea of continuous deformation.
This module will concentrate on constructing the most basic family of such invariants, homology groups , and the applications of these homology groups.
Course on Homotopy Theory (first semester 2012/2013)
The starting point will be simplicial complexes and simplicial homology. An n-simplex is the n-dimensional generalisation of a triangle in the plane. A simplicial complex is a topological space which can be decomposed as a union of simplices.
The simplicial homology depends on the way these simplices fit together to form the given space. Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex.
For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes. Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space.
 Homotopy type theory: the logic of space
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Homotopic Topology by Anatoly Fomenko Dmitry Fuchs
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