Simplicial Sets_ and Their Application to Computing Homology

					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
Example Simplification
Another Simplification
    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
What has Trivial Homology?
V   E     F   0    1   2
3   3     1   0     0    0
2   3     1   0     1    0
1   3     1   0     2    0
2   2     1   0     0    0
1   2     1   0     1    0
1   1     1   0     0    0
1   0     1   0     0    1
2   1     0   0     0    0
1   1     0   0     1    0
Example From Before
   Makes Sense
New Example: Torus
        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
Torus (Revisited)
      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

				
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posted:8/15/2011
language:English
pages:23