Homology of Narrow Posets

Gabriel Drew Carroll
2001 Intel STS Third Place Winner
Oakland Technical High School, Oakland, CA

(This is an abridged version of Gabriel’s report.  The original report included Summary, Abstract, Introduction, Statement of Results, Preliminary Results, Proofs of Main Results, Further Study, and Acknowledgments sections.)

Introduction

Partially ordered sets (posets) are important throughout mathematics.  A poset is a set with an ordering relation on some (not necessarily all) pairs of elements.  If every two elements are comparable, the poset is said to be totally ordered.  A simplicial complex is a geometrical object obtained by gluing together simplices (triangles and their analogues in other dimensions).  Given a simplicial complex, one can talk about its homology groups, which are algebraic structures describing — in some sense — its topological complexity.

Given a poset, we may form a simplicial complex by letting the elements become points and letting each totally ordered subset become a simplex.  It is common to study the poset in terms of the homology groups of this complex.  While several previous results exist that may be used to estimate the complexities of these groups, none is expressed in terms of an easily described numerical property of the poset.  We close this gap by obtaining an estimate in terms of the poset’s width, defined to be the maximum number of mutually incomparable elements.

Statement of results

Let P denote the order complex of the poset P; let H_d denote the d-th reduced homology functor.  For any abelian group G, let g(G) be the minimal cardinality of a generating set for G. Then:

Theorem 1: Let P be a poset of width w > 1.

In particular, if P is an infinite poset of finite width, then all its homology groups are finitely generated, and the sum above converges.

This is actually a special case of a more general theorem.  To state it, we introduce the notion of a strong gate (defined in the full paper).  Then:

Theorem 2: Let P be a poset of width at most w (w > 1), and suppose that P contains r strong gates (r > 0) of cardinalities v₁, v₂, …, v_r, such that any two elements in different strong gates are comparable.  Then

We also construct infinitely many examples to show that these bounds are achievable.

First suppose P is finite; we prove the result by induction on its cardinality.

Case 1: P is connected.  In this case, we express P as a union of at most v cones, which have zero homology.  We relate the homology of P to the homologies of various unions and intersections of these cones, for which bounds are obtained by the induction hypothesis, and the result follows.

Case 2: P is disconnected.  In this case, we apply the induction hypothesis to the various components and add.

Next, a lemma: if P is an infinite poset of finite width, every homology group is finitely generated.  The proof is by induction on the homology dimension d; as before, we express the homology of the whole poset in terms of lower-dimensional homologies of unions and intersections of finitely many cones, for which we can apply the induction hypothesis.

Finally, to prove Theorem 2 in the infinite case, we suppose some counterexample exists.  With the aid of the lemma, we find a finite subposet which also violates the theorem, and this is already known to be a contradiction.

We also prove Corollary 1.  In the finite case, it readily holds by letting w go to infinity in Theorem 2.  If P is infinite, we again show that any poset violating the corollary must have some finite subposet violating the corollary, which is impossible.

Further study

These results lay the groundwork for studying the connection between poset width and homology. Many areas remain to be examined, such as torsion in homology groups, simplification of the hypotheses of our results, and finiteness constraints for infinite posets' homology. It is also worth wondering whether applications of this work to "naturally occurring" posets could be meaningful; while none are known, there is hope.