Configuration Spaces in Algebraic Topology: Lecture 1
نویسنده
چکیده
Perspective (Invariants). The homotopy type of a fixed configuration space is a homeomorphism invariant of the background space, and, in the case of a manifold, it turns out that these invariants remember a rather large amount of information. A simple-minded example is provided by Euclidean spaces of different dimension; indeed, as we will see, there is a homotopy equivalence B2(R) ' B2(R) if and only if m = n. In other words, configuration spaces are sensitive to the dimension of a manifold. A somewhat more sophisticated example is provided by the fact that B2(T \pt) 6' B2(R\S), which can be shown by a homology calculation. Note that T 2 \ pt and R \ S have the same dimension and homotopy type, having S ∨ S as a common deformation retract. On the other hand, (T 2 \ pt) ∼= T 2 6' S ∨ S ∨ S ' (R \ S), so we might conclude from this example that configuration spaces are sensitive to the proper homotopy type of a manifold. In order to discuss the most striking illustration of the sensitivity of configuration spaces, we recall that the Lens spaces are a family of compact 3-manifolds given by
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تاریخ انتشار 2017