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

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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