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 = { K 0, K 1, …, K n, …}, together with functions d i : K n  K n-1 s i : K n  K n+1 for each 0  i  n

Simplicial Identities d i d k = d k-1 d i for i < k d i s k = s k-1 d i for i < k = identity for i = j, j+1 = s k d i-1 for i > k + 1 s i s k = s k+1 s i 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. K n = { n-simplices } d i = delete vertex in position i s i = repeat vertex in position i

Homology of Simplicial Set Chain complexes are the free abelian groups on the n-simplices Boundary operator:    (-1) i d i Degenerate (x = s i 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? VEF 00 11 2

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(H n (X)) = d(H n (A)) + d(H n (X/A)) + d(ker i n-1 * ) - d(ker i n * ) 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