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May, 2019
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Important Spaces and Operation on Them

In topology, we study topological spaces and continuous maps between them. Which spaces should we begin to study? There are many wierd or pathological spaces, which are chosen as examples in textbooks to show the intuition of the reader is false.

What Poincaré wanted to study in “Analysis Situs” [ Poi96 ] is the topology of manifolds . The category of manifolds is not a good place to play. Possible operations are quite limited. In fact, Poincaré’s first attempt to define the “homology group” as the set of the equivalence classes of submanifolds under the relation “homologous” by taking the union of two manifolds and to define the addition in homology by taking unions of submanifolds caused a lot of discussions and criticism. The union of submanifolds is not necessarily a manifold.

In order to overcome this difficulty, Poincaré used triangulations. Namely he defined homology groups for simplicial complexes . The world of simplicial complexes is not large enough. We usually work in the category of CW-complexes or simplicial sets . And there are some more attempts to construct convinient categories for doing algebraic topology .