Créer une présentation
Télécharger la présentation

Télécharger la présentation
## Simplicial Sets, and Their Application to Computing Homology

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Simplicial Sets, and Their Application to Computing Homology**Patrick Perry November 27, 2002**Simplicial Sets: An Overview**• A less restrictive framework for representing a topological space • Combinatorial Structure • Can be derived from a simplicial complex • Makes topological simplification easier • Possibly a good algorithm for Homology computation**Motivation**• If X is a topological space, and A is a contractible subspace of X, then the quotient map X X/A is a homotopy equivalence • Any n-simplex of a simplicial complex is contractible**Geometry Is Not Preserved**• Collapsing a simplex to a point distorts the geometry • After a series of topological simplifications, a complex may have drastically different geometry • Does not matter for homology computation**Cannot use a Simplicial Complex!**• Bizarre simplices arrise: face with no edges, edge bounded by only one point • Need a new object to represent these pseudo-simplices • Need supporting theory to justify the representation**Simplicial Sets**• A Simplicial Set is a sequence of sets K = { K0, K1, …, Kn, …}, together with functions di : Kn Kn-1 si : Kn Kn+1 for each 0 i n**Simplicial Identities**• didk = dk-1di for i < k • disk = sk-1di for i < k = identity for i = j, j+1 = skdi-1 for i > k + 1 • sisk = sk+1si for i k**Simplicial Complexes as Simplicial Sets**• A simplicial set can be constructed from a simplicial complex as follows: Order the vertices of the complex. Kn = { n-simplices } di = delete vertex in position i si = repeat vertex in position i**Homology of Simplicial Set**• Chain complexes are the free abelian groups on the n-simplices • Boundary operator: (-1)i di • Degenerate (x = si y) complexes are 0 • Homology of Simplicial Set is the same as the homology of the simplicial complex**Bizarre Simplices are OK**• Simplicial sets allow us to have an n-simplex with fewer faces than an n-simplex from a simplicial complex • Our bizarre collapses make sense in the Simplicial Set world**End Result for Torus**• We have eliminated 8 faces, 16 edges, and 8 vertices • Cannot simplify any further without affecting homology**Benefit of Simplicial Set**• More flexibility in what we are allowed to do to a complex • Linear-time algorithm to reduce the size of a complex • Can use Gaussian Elimination to compute Homology of simplified complex**Can We Simplify Further?**• What about (X X/A) + bookkeeping?**Bookkeeping**• Using Long Exact Sequence, we can figure out how to simplify further: d(Hn(X)) = d(Hn(A)) + d(Hn(X/A)) + d(ker in-1*) - d(ker in*) • If i* is injective, bookkeeping is easy**Collapsing the Torus to a Point**• Inclusion map on Homology is injecive in each simplification • = (0, 0, 0) + (0, 1, 0) + (0, 1, 0) + (0, 0, 1) = (0, 2, 1)**Good News**• Computation of ker i* is local • Potentially compute homology in O(n TIME(ker i* ))**Conclusion**• A less restrictive combinatorial framework for representing a topological space • Can be derived from a simplicial complex • Makes topological simplification easier • Possibly a good algorithm for Homology computation