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Simplicial Sets, and Their Application to Computing Homology

This document explores simplicial sets as a less restrictive and more flexible framework for representing topological spaces. It highlights the advantages of using simplicial sets for topological simplification and homology computation, emphasizing that these sets allow for a broader range of configurations than traditional simplicial complexes. By eliminating certain faces, edges, and vertices, simplicial sets facilitate efficient algorithms for homology computation while maintaining the integrity of essential topological features. The study presents methods for constructing simplicial sets from simplicial complexes and discusses implications for computational homology.

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Simplicial Sets, and Their Application to Computing Homology

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  1. Simplicial Sets, and Their Application to Computing Homology Patrick Perry November 27, 2002

  2. 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

  3. 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

  4. Example Simplification

  5. Another Simplification

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. What has Trivial Homology?

  14. Example From Before Makes Sense

  15. New Example: Torus

  16. End Result for Torus • We have eliminated 8 faces, 16 edges, and 8 vertices • Cannot simplify any further without affecting homology

  17. 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

  18. Can We Simplify Further? • What about (X  X/A) + bookkeeping?

  19. 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

  20. Torus (Revisited)

  21. 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)

  22. Good News • Computation of ker i* is local • Potentially compute homology in O(n TIME(ker i* ))

  23. 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

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