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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
= identity
= skdi-1
• sisk = sk+1si
for i < k
for i = j, j+1
for i > k + 1
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 nsimplex from a simplicial complex
• Our bizarre collapses make sense in the
Simplicial Set world
What has Trivial Homology?
V
3
2
1
2
1
1
1
2
1
E
3
3
3
2
2
1
0
1
1
F
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
0
1
0
0
0
1
2
0
0
0
0
0
0
1
0
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