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homeotopy∗
juanman†
2013-03-21 20:04:59
Let X be a topological Hausdorff space. Let Homeo(X) be the group of
homeomorphisms X → X, which can be also turn into a topological space by
means of the compact-open topology. And let πk be the k-th homotopy group
functor.
Then the k-th homeotopy is defined as:
Hk (X) = πk (Homeo(X))
that is, the group of homotopy classes of maps S k → Homeo(X). Which is
different from πk (X), the group of homotopy classes of maps S k → X.
One important result for any low dimensional topologist is that for a surface
F
H0 (F ) = Out(π1 (F ))
which is the F ’s extended mapping class group.
Reference
G.S. McCarty, Homeotopy groups, Trans. A.M.S. 106(1963)293-304.
∗ hHomeotopyi created: h2013-03-21i by: hjuanmani version: h37642i Privacy setting:
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