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Date: 6-7-2021
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A handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a zip around each puncture travelling in opposite directions, pulling the edges of the zips together, and then zipping up.
Handles are to manifolds as cells are to CW-complexes. If is a manifold together with a -sphere embedded in its boundary with a trivial tubular neighborhood, we attach a -handle to by gluing the tubular neighborhood of the -sphere to the tubular neighborhood of the standard -sphere in the dim()-dimensional disk. In this way, attaching a -handle is essentially just the process of attaching a fattened-up -disk to along the -sphere . The embedded disk in this new manifold is called the -handle in the union of and the handle.
Dyck's theorem states that handles and cross-handles are equivalent in the presence of a cross-cap.
REFERENCES:
Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof." Amer. Math. Monthly 106, 393-399, 1999.
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