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Handle


HandleZip

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 M is a manifold together with a (k-1)-sphere S^(k-1) embedded in its boundary with a trivial tubular neighborhood, we attach a k-handle to M by gluing the tubular neighborhood of the (k-1)-sphere S^(k-1) to the tubular neighborhood of the standard (k-1)-sphere S^(k-1) in the dim(M)-dimensional disk. In this way, attaching a k-handle is essentially just the process of attaching a fattened-up k-disk to M along the (k-1)-sphere S^(k-1). The embedded disk in this new manifold is called the k-handle in the union of M and the handle.

Dyck's theorem states that handles and cross-handles are equivalent in the presence of a cross-cap.


See also

Cap, Classification Theorem of Surfaces, Cross-Cap, Cross-Handle, Dyck's Theorem, Handlebody, Surgery, Tubular Neighborhood

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References

Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof." Amer. Math. Monthly 106, 393-399, 1999.

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Handle

Cite this as:

Weisstein, Eric W. "Handle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Handle.html

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