Analytic Subdivision Invariants
This paper introduces an inner product on chain complexes of finite simplicial complexes that is well-adapted to the harmonic study of subdivisions. Its definition utilizes a decomposition of the chain spaces that suggests a sequence of subdivision invariants which we show do not all vanish for non-trivial subdivisions. We exhibit a combinatorial lower bound for these invariants and provide an effective algorithm for their computation. Unfortunately, these invariants cannot distinguish every subdivision nor do they necessarily increase over successive subdivisions.
Let be a fixed, finite simplicial complex and subdivisions of . We say that are isomorphic provided there is a simplicial isomorphism between them. A natural question is to determine whether any two given subdivisions are equivalent. Certainly, one can enumerate all the possible simplicial maps between their vertex schemes, but to the author’s knowledge there is not yet a more effective way. As in topology, one could begin by checking invariants. By a (real-valued) subdivision invariant, we mean an assignment of real numbers such that if , then . Equivalently, is a function on the poset of subdivisions of which descends to the quotient poset of simplicial isomorphism classes. The most classical examples are the face numbers, but these are easily seen to be insufficient; for example, see Figure 1.
This paper examines a sequence of analytic subdivision invariants derived from inner product structures on the chain spaces. The use of inner product structures to study finite simplicial complexes was initiated by Eckmann in  (see  for more recent applications). The definition of the invariants is based on a particular decomposition of the chain spaces. Though each chain space has a canonical inner product given by the simplices, this decomposition can also be used to associate to each subdivision an inner product on the chain spaces which reflects the harmonic theory on . The following is an amalgamation of Theorem 2.7 and Corollary 2.11:
Let be a subdivision of a simplicial complex . There exists subspaces such that
If is an inner product on , then this decomposition defines an inner product on which satisfies
where are the Laplacians relative respectively.
This allows the harmonic theory of the standard inner products on to be compared on just one complex. In particular, one can study the norm of the differences of harmonic representatives. The connections between harmonic theory and subdivisions will be the subject of a future paper. Here, the above decomposition is used to define the following sequence of numbers: for each non-negative integer , consider
which measures the extent the subspace fails to be -invariant relative the adjoint with respect to the standard inner product on the chain space . The numbers are subdivision invariants (Theorem 3.4) and do not all vanish for non-trivial subdivisions (Theorem 4.13). The latter claim follows from the following combinatorial bound for these invariants. To state this bound, we introduce the following definition which will be useful for geometric arguments throughout this paper:
Let be a subdivision of simplicial complexes. A -party of is a union of -simplices of which coincides with the image of a -simplex of under the simplex map . The -simplices of which constitute a -party are called its members.
We sometimes find it expedient to think of parties as linear combinations in the chain space.
Consider the -complex which is the union of the two intervals and . Suppose the former is subdivided at . Then, the resulting complex has two -parties: and .
Let be a subdivision of a simplicial complex and any -simplex of . Write for the number of incident -simplices and for the number of -faces of these incident -simplices which are supported on singly represented parties, i.e. parties that support no two of these -faces. Here is the combinatorial lower bound (Proposition 4.6):
Let be a subdivision of a simplicial complex, and suppose that the number of -simplices has increased. Pick any -simplex not supported on any -parties. Then,
These invariants admit a geometric interpretation as a constrained optimization of a quadratic functional over a conic intersection locus. An effective algorithm for their computation (illustrated in Example 5.16) is provided and the codimension one value is deduced for elementary stellar subdivision of an isolated simplex (Proposition 6.10) or along an interior simplex (Proposition 6.16). Using the algorithm, one can compute that the sequence distinguishes between the subdivisions in Figure 1 whereas the face numbers could not. However, they are still insufficient to distinguish between any two subdivisions of a common simplicial complex (Remark 6.11) and do not necessarily increase with successive subdivision (Example 4.23).
The sections are organized as follows: Section 2 introduces an inner product well adapted for harmonic theory on subdivisions via a particular decomposition of the chain spaces. The sequence of subdivision invariants studied in this paper is introduced in Section 3 and a combinatorial lower bound is given in Section 4. Section 5 provides the geometric interpretation and an effective algorithm for their computation. This interpretation is used in Section 6 to obtain the codimension one values for an elementary stellar subdivision in certain cases. Finally, the appendix (Section 7) provides background on the Laplacian for chain complexes.
2. Subdivisions and Inner Products
Let be a subdivision of simplicial complexes. Suppose are inner products on the chain spaces respectively. Because the induced map on chain spaces is injective, one can readily define an inner product on such that . Thus, the combinatorial structure of the unsubdivision is encoded at the level of the chain spaces . The choice of such that its pullback is is in general not unique. In this section, we will show that there is a canonical choice which further satisfies the property that the chain map and the Laplacians commute:
where are the Laplacians relative respectively. This will allow us to conduct Laplacian calculations on the subdivided complex alone. Hopefully, questions about a subdivision pair may now be adequately translated into a question about a pair of inner products on a single vector space. To define the canonical choice, we decompose into convenient summands.
A sufficient, but not necessary condition for Equation (2.1) is that and commute.
Suppose . The maps commute iff the -orthogonal complement of is -invariant.
First, suppose commute. Let and any chain. Then,
so that is -invariant.
Conversely, suppose the -orthogonal complement of is -invariant. For any chain we write for and . Then, for any chain ,
so that . Also, because we have
so that . Thus, for all chains and so that . ∎
Note that -invariance implies that the induced map
is injective, though this result is true more generally:
Let be a subdivision of simplicial complexes. The map
induced by the boundary map is injective for all .
Consider the diagram
Suppose for some chain . Since , the isomorphism implies that . Hence, there is a -chain such that . We then have so that . ∎
Note that we do not have . This is the observation behind the subdivision invariants to be introduced in Section 3.
Now we can write the desired decomposition:
Let be a subdivision of simplicial complexes, and equip with an inner product. Then, there exist subspaces such that
Let be a subdivision of simplicial complexes, and suppose the chain spaces are equipped with inner products respectively. There exists an inner product on such that the associated Laplacians and inclusions commute:
We define an inner product on by setting the summands
mutually orthogonal while using the induced inner product from on both but on . By construction and the -complement of is -invariant so that Proposition 2.2 implies that commute. Hence, . ∎
We will call the inner product defined in the preceding argument the canonical inner product for the morphism of inner product spaces. When we use the standard inner products given by the simplices, we will say that is the canonical inner product for subdivision .
The following relations summarize the behavior of the decomposition with respect to the boundary map and its adjoint relative either inner product or :
The constructions in this section are algebraic and thus valid in the setting of chain complexes of inner product spaces and injective morphisms which induce isomorphisms on homology.
3. An Analytic Approach to Subdivision Complexity
In this section, we introduce a subdivision invariant based on the canonical decomposition (2.8) for a subdivision. The original context was the study of the norm of the difference of cycle representatives of a fixed homology class (see below).
Let be an inner product space, and subspaces. We write for the quantity
where and , and we will omit the subscript when using the standard inner product.
Recall that by Theorem 2.7, for each subdivision , there is an associated canonical decomposition relative the standard inner product on ,
For each non-negative integer , define
The values are invariants of the subdivision.
Let be a subdivision and a simplicial isomorphism. Because is a dimension-preserving map between simplices (in fact, just re-labeling simplices), the induced map is an isometry and we have . Thus,
so that the values are identical for isomorphic subdivisions of a given complex. ∎
Evidently, , and if is the trivial subdivision, then for all . We will see subsequently (Theorem 4.13) that these invariants do not all vanish for non-trivial subdivisions.
These quantities arise somewhat naturally in the study of the norm of the difference between certain cycle representatives of a fixed homology class. Again, let be a subdivision and endow with an inner product . Fixing a homology class for , let be the harmonic representative in relative and the inclusion under of any cycle representative in . Then, from the relations
where , we see that
The subspace inherits an induced inner product and has a Hodge decomposition:
because by injectivity of . Then,
Since these two subspaces intersect non-trivially, so that
However, from the canonical decomposition, we know that so that taking recovers a potentially interesting extremization. In fact, since , we know that
so that is a measure of the failure of to be closed under the adjoint relative . Alternatively, it is a measure of how and project onto each other and hence the “angle” between them.
One could have used instead the canonical inner product associated to the subdivision, and similarly obtain where , the chain is -harmonic, and is an arbitrary cycle. Again but because , there is no interesting extremization in this case. Indeed, is by construction -invariant since the pullback of by the subdivision map is the standard inner product on , as if everything were done on alone. More easily, this also follows because commute.
If in our analysis, we restricted to be -harmonic, then one can show that the subspaces and intersect trivially and that defines another subdivision invariant. Analysis of this invariant, though only useful in cases of non-trivial homology, will be the subject of a future paper.
4. Properties of the Invariants
In this section, we derive a combinatorial lower bound on the invariants (Theorem 4.6) which will show that these invariants do not all vanish for non-trivial subdivisions (Theorem 4.13). We also discuss the difficulties in estimating over successive subdivisions and provide an example where does not increase with successive subdivision (Example 4.23).
4.1. Combinatorial Lower Bound for Invariants
The central result of this subsection is a combinatorial lower bound for the invariants (Theorem 4.6). This bound implies that for any non-trivial subdivision, there is a (Theorem 4.13). We also present explicit bases for the subspaces (Propositions 4.14 and 4.15). We begin by computing the dimension of .
Let denote the number of -simplices of a simplicial complex. If is a subdivision, then
In particular, the number of new vertices.
Since , the preceding proposition implies the inequality for odd and the reverse inequality for even.
If is a -simplex not supported on any -party, then .
By the relations in (2.12), , so we need only show that is perpendicular to . We present two arguments, the first analytic, the second geometric.
Analytic argument: For any , we have by hypothesis. Thus, is perpendicular to and hence an element of .
Geometric argument: First, the -chain is supported on -simplices incident to . Now, either the -simplex is supported on a -party or not. If not, then is not supported on any -party and hence perpendicular to . If so, incident -simplices are of two types depending on whether they are members of the necessarily non-singular -party. Now, the party members are -dimensional simplices of a subdivision of a -simplex. Hence, there are only two such, namely those determined by having as their shared boundary, and their contribution in is thus perpendicular to . The remaining incident -simplices are not supported on , and we conclude that is perpendicular to . ∎
Let be a subdivision of a simplicial complex and any -simplex of . Write for the number of incident -simplices and for the number of -faces of these incident -simplices which are supported on singly represented parties, i.e. parties that support no two of these -faces.
Let be a subdivision of a simplicial complex and suppose that the number of -simplices has increased. Pick any -simplex not supported on any -parties. Then,
Since the number of -simplices has increased, there is a non-singular -party. Any two party members will share a -simplex not supported on any -party. By the Lemma 4.5, we have so that extremization over the subspace gives a lower bound
Now, where the latter sum has terms, the number of -faces of other than . Then, and where is the projection onto . The latter equality reflects the fact that if two members of the same -party are present in (and by the geometry of simplices, there can be at most two), then their contribution is orthogonal to . Thus,
and the inequality follows. ∎
Let be an elementary stellar subdivision with being the new vertex. Write for the number of vertices in its link . Then, and since , we have so that
In particular, all elementary stellar subdivisions of an isolated -simplex have the same . This equality offers a partial combinatorial description of the analytic invariant . The invariant also gives a lower bound on , but this is useful only when .
For the elementary stellar subdivision in Figure 4, choosing the edge , we have and (because the contribution in by the edges in the base -party will be perpendicular to ) so that
and by the preceding example.
Consider the codimension one invariant for a subdivision of a combinatorial -manifold. For any -simplex , we have . Hence,
where the maximum is taken over all codimension one simplices . When is known, this gives a lower bound for .
If the number of -simplices has changed due to a subdivision, then . In particular, for non-trivial subdivisions of pure simplicial complexes, we have in all positive codimensions.
This follows from the positivity of the lower bound under the hypothesis. Subdivisions of pure simplicial complexes increase the number of simplices in any dimension where simplices were already present. ∎
In the remainder of this subsection, we present explicit descriptions for the subspaces .
Let be the set of new vertices. Then, is a basis for .
Since the new vertices are not supported on -parties, by Lemma 4.5, we have .
Next, we show that the -chains are linearly independent. Note that we need only show that is not orthogonal to any non-zero subspace of the span of the new vertices. Let and suppose . Let be any -chain that is a directed path from some original vertex to . Then, so that the -chains are linearly independent. Since by Proposition 4.1, is the number of new vertices, we conclude that is a basis for . ∎
Now, we describe by exhibiting a basis where are particular -simplices. Note that
where is the number of new vertices. We will specify number of -simplices, and the remaining edges will then be our desired . First, write where is the number of new vertices supported on -parties and the number not supported. Then, the number of -simplices supported on -parties is . For each -party with , choose a spanning tree for the (new) interior vertices, and pick any edge supported on that party connecting the tree to an old vertex, which we will call the anchor for the tree. The number of chosen edges in each -party is then the number of new vertices supported in its interior, and hence edges have been specified by the trees and anchors. Together with edges in -parties, we have now specified edges. The remaining number of edges are our .
Let be edges as described above. Then, is a basis for .
By construction, such edges are not supported on -parties. Hence, we conclude by Lemma 4.5 that .
To show that the set is linearly independent, we proceed as we did for , namely we will show that no non-trivial subspace of the span of is orthogonal to . first, note that for any such edge , either both endpoints are on the same spanning tree or not.
If so, let denote the set of edges in the tree as well as the edge connection to the anchor. Then determines a -cycle in conjunction with the spanning tree. This cycle is supported on the -party defining the tree, and since -parties are topologically trivial, the cycle is a boundary. Hence, there is a -chain supported on the -party for which .
If the endpoints are on different spanning trees, let denote there respective edges and anchor connections, and the party of minimal dimension supporting both . (The party exists because is not supported on any -parties.) On each spanning tree, there is a (unique) path from the supported vertex of to the respective anchor. The anchors themselves are connected by -parties because they are old vertices, and this determines a -cycle supported on . Again, topological triviality implies the existence of a supported -chain such that now . Hence for each edge , there is a -chain such that so that no non-trivial subspace of is orthogonal to . ∎
4.2. Successive Subdivisions
In this subsection we describe the difficulty in estimating the invariants over successive subdivisions and provide an example (4.23) that shows need not increase with successive subdivision.
Let be a sequence of subdivisions. The fundamental observation is that the canonical inner product for the entire subdivision differs from that obtained via sequential application. With respect to the subdivision we have
whereas for individually, we have
where all the orthogonality relations are with respect to the standard inner product on the appropriate ambient space. The crux of the discrepancy is that inclusion by of into does not preserve the orthogonality relation originally present in . Hence, we only have
In particular, may not be orthogonal to the subspace because the complements are determined by the inner product on the intermediate , not by the standard inner product on . Moreover, neither do we necessarily have . To obtain a decomposition respecting the subspaces, we would need to use the pullback of the standard inner product on . Then, .
Let and . Then, because . Thus, we may refine the decomposition (4.18) as
Let , be a subdivision by adding a vertex at , and another vertex at , which we summarize as follows:
Numbering the edges in increasing order within each complex, we obtain the following relations in the top dimension:
but so that is not orthogonal to .
This example also shows that may not be a subset of . If we label the vertices in increasing order, then
and one readily checks that .
where the subscripts refer to the standard inner products on respectively. The following example shows that not all the invariants necessarily increase with successive subdivision.
We show that, unfortunately, the invariant does not necessarily increase over successive subdivisions. Consider a simplicial complex under two successive elementary stellar subdivisions. Let be the added vertices in that order. Suppose that their star neighborhoods are disjoint in the final complex so that all the vertices in their links are old vertices. Let be the number of vertices in their respective links. Then, where and likewise for . By Proposition 4.14, where is the projection onto the inclusion of original chains. Set
Then, for the combined subdivision is the maximum of over all real , which we seek by equivalently maximizing its square, the function