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
What has Trivial Homology?
V E F 0 1 2
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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
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)
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