Collapsing along monotone poset maps

We introduce the notion of nonevasive reduction, and show that for any monotone poset map $\phi:P\to P$, the simplicial complex $\Delta(P)$ {\tt NE}-reduces to $\Delta(Q)$, for any $Q\supseteq{\text{\rm Fix}}\phi$. As a corollary, we prove that for any order-preserving map $\phi:P\to P$ satisfying $\phi(x)\geq x$, for any $x\in P$, the simplicial complex $\Delta(P)$ collapses to $\Delta(\phi(P))$. We also obtain a generalization of Crapo's closure theorem.


Order complexes, collapsing and NE-reduction.
For a poset P we let ∆(P ) denote its nerve: the simplicial complex whose simplices are all chains of P . For a simplicial complex X we let V (X) denote the set of its vertices.
An elementary collapse in a simplicial complex X is a removal of two open simplices σ and τ from X, such that dim σ = dim τ + 1, and σ is the only simplex of X, different from τ itself, which contains the simplex τ in its closure.
When Y is a subcomplex of X, we say that X collapses onto Y if there exists a sequence of elementary collapses leading from X to Y ; in this case we write X ց Y (or, equivalently, Y ր X).
(1) A finite nonempty simplicial complex X is called nonevasive if either X is a point, or, inductively, there exists a vertex v of X, such that both X \ {v} and lk X v are nonevasive.
(2) For two nonempty simplicial complexes X and Y we write X ց NE Y (or, equivalently, Y ր NE X), if there exists a sequence X = A 1 ⊃ A 2 ⊃ · · · ⊃ A t = Y , such that for all i ∈ {1, . . . , t − 1} we can write V ( We recall, that the notion of nonevasive simplicial complexes was introduced in [KSS], and was initially motivated by the complexity-theoretic considerations. For further connections to topology and more facts on nonevasiveness we refer to [Ku90,We99]. Recently an interesting connection has been established between Discrete Morse theory and evasiveness, the standard references are [Fo00,Fo98].
Several classes of simplicial complexes are known to be nonevasive. Perhaps the simplest example is provided by the fact that all cones are nonevasive. A more complicated family of nonevasive simplicial complexes is obtained by taking the order complexes of the noncomplemented lattices, see [Ko98].
In the situation described in Definition 1.1(2), we say that the simplicial complex X NE-reduces to its subcomplex Y . The following facts about NEreduction are useful for our arguments.
Fact 1. If X 1 and X 2 are simplicial complexes, such that X 1 ց NE X 2 , and Y is an arbitrary simplicial complex, then X 1 * Y ց NE X 2 * Y .
Here the symbol * denotes the simplicial join of two simplicial complexes, see [Mu84]. That X 1 * Y ց NE X 2 * Y follows by induction from the facts that if v is any vertex of a simplicial complex X, then we have lk X To NE-reduce X 1 * Y to X 2 * Y , simply take the sequence of vertices x 1 , . . . , x t ∈ V (X 1 ) which NE-reduces X 1 to X 2 . We have lk X 1 * Y (x 1 ) = (lk X 1 (x 1 )) * Y . In turn, the simplicial complex (lk X 1 (x 1 )) * Y is nonevasive: this is seen by induction on the number of vertices of the first factor, with the base given by the fact that all cones are nonevasive. Removing x 1 from X 1 * Y yields the simplicial complex (X 1 \ {x 1 }) * Y , hence, continuing in this way, we will NE-reduce X 1 * Y all the way to X 2 * Y .
Fact 2. The reduction X ց NE Y implies X ց Y , which in turn implies that Y is a strong deformation retract of X.

Monotone poset maps.
Next we define a class of maps which are particularly suitable for our purposes.
Definition 2.1. Let P be a poset. An order-preserving map ϕ : P → P is called a monotone map, if for any x ∈ P either x ≥ ϕ(x) or x ≤ ϕ(x).
If x ≥ ϕ(x) for all x ∈ P , then we call ϕ a decreasing map, analogously, if x ≤ ϕ(x) for all x ∈ P , then we call ϕ an increasing map.
We remark here the fact that while a composition of two decreasing maps is again a decreasing map, and, in the same way, a composition of two increasing maps is again an increasing map, the composition of two monotone maps is not necessarily a monotone map. On the other hand, any power of a monotone map is again monotone. Indeed, let ϕ : P → P be monotone, let x ∈ P , and say x ≤ ϕ(x). Since ϕ is The following proposition shows that monotone maps have a canonical decomposition in terms of increasing and descreasing maps.
Proposition 2.2. Let P be a poset, and let ϕ : P → P be a monotone map. There exist unique maps α, β : P → P , such that • α is an increasing map, whereas β is a decreasing map; Proof. Set Thus (2.1) gives ϕ(x) ∈ Fix α, and we conclude that α(β(x)) = α(ϕ(x)) = ϕ(x). To see that α is an increasing map, we just need to see that it is orderpreserving. Since α either fixes an element or maps it to a larger one, the only situation which needs to be considered is when x, y ∈ P , x < y, and α(x) = ϕ(x). However, under these conditions we have α(y) ≥ ϕ(y) ≥ ϕ(x) = α(x), and so α is order-preserving. That β is a decreasing map can be seen analogously. Finally, the uniqueness follows from the fact that each x ∈ P must be fixed by either α or β, and the value ϕ(x) determines which one will fix x.

The main theorem and implications.
Prior to this work, it has been known that a monotone map ϕ : P → P induces a homotopy equivalence between ∆(P ) and ∆(ϕ(P )), see [Bj95,Corollary 10.17]. It was also proved in [Bj95] that if the map ϕ satisfies the additional condition ϕ 2 = ϕ, then ϕ induces a strong deformation retraction from ∆(P ) to ∆(ϕ(P )).
The latter result was strengthened in [Ko04, Theorem 2.1], where it was shown that, whenever ϕ is an ascending (or descending) closure operator, ∆(P ) collapses onto ∆(ϕ(P )). There this fact was used to analyze the effect of the folding operation on the corresponding Hom complexes, see also [BK03a,BK03b,BK04,Ko05].
The next theorem strengthens and generalizes the results from [Bj95] and [Ko04].
Theorem 3.1. Let P be a poset, and let ϕ : P → P be a monotone map. Assume P ⊇ Q ⊇ Fix ϕ, P \ Q is finite, and, for every x ∈ P \ Q, P <x ∪ P >x is finite, then ∆(P ) ց NE ∆(Q), in particular, ∆(P ) collapses onto ∆(Q).

Remarks.
1) Note that when P is finite, the conditions of the Theorem 3.1 simply reduce to: P ⊇ Q ⊇ Fix ϕ.
2) Under conditions of Theorem 3.1, the simplicial complex ∆(P ) collapses onto the simplicial complex ∆(Q), in particular, the complexes ∆(P ) and ∆(Q) have the same simple homotopy type, see [Co68].
3) Under conditions of Theorem 3.1, the topological space ∆(Q) is a strong deformation retract of the topological space ∆(P ). 4) Any poset Q satisfying P ⊇ Q ⊇ ϕ(P ) will also satisfy P ⊇ Q ⊇ Fix ϕ, hence Theorem 3.1 will apply. In particular, for finite P , we have the following corollary: Corollary 3.2. For any poset P , and for any monotone map ϕ : P → P , satisfying conditions of Theorem 3.1, we have ∆(P ) ց NE ∆(ϕ(P )).
It is easy to prove Theorem 3.1, once the following auxiliary result is established.
Proof. Since the expression ∆(P <x ) * ∆(P >x ) is symmetric with respect to inverting the partial order of P , it is enough, without loss of generality, to only consider the case ϕ(x) < x. Let us show that in this case ∆(P <x ) is nonevasive. We proceed by induction on |P <x |. If |P <x | = 1, then the statement is clear, so assume |P <x | ≥ 2.
Let ψ : P <x → P <x denote the restriction of ϕ. It is easy to see that ψ is a monotone map of P <x . To verify that ∆(P <x ) ց NE ∆(P ≤ϕ(x) ), order the elements of P <x \ P ≤ϕ(x) following an arbitrary linear extension in the decreasing order, say P <x \ P ≤ϕ(x) = {a 1 , . . . , a t }, and a i < a j , for i < j. By the choice of the order of a i 's, we have P <a i = P i <a i , where P i = P \ {a 1 , . . . , a i−1 }. Therefore, by the induction assumption, ∆(P <a i ) is nonevasive for all 1 ≤ i ≤ t, and we have ∆(P <x ) = ∆(P 1 <x ) ց NE ∆(P 2 <x ) ց NE . . . ց NE ∆(P t+1 <x ) = ∆(P ≤ϕ(x) ). On the other hand, ∆(P ≤ϕ(x) ) is a cone, hence it is nonevasive, and therefore ∆(P <x ) is nonevasive as well. It follows that ∆(P <x ) * ∆(P >x ) is nonevasive.
Proof of the Theorem 3.1. The proof is by induction on |P \ Q|. The statement is trivial when |P \ Q| = 0, so assume |P \ Q| ≥ 1.
To start with, we replace the monotone map ϕ with a monotone map γ satisfying γ(P ) ⊆ Q and Fix γ = Fix ϕ. To achieve that objective we can set γ := ϕ N , where N = |P \ Q|. With this choice of γ, the inclusion γ(P ) ⊆ Q follows from the assumption that Fix ϕ ⊆ Q, since Fix γ = γ(P ).
On the enumerative side, we obtain the following generalization of the Crapo's Closure Theorem from 1968, see [Cr68, Theorem 1].
Before we give the proof, recall the following convention: whenever P is a poset with0 and1, we letP denote P \ {0,1}.

NE-reduction and collapses
The NE-reduction can be used to define an interesting equivalence relation on the set of all simplicial complexes.
Definition 4.1. Let X and Y be simplicial complexes. Recursively, we say that X ≃ NE Y if X ց NE Y , or Y ր NE X, or if there exists a simplicial complex Z, such that X ≃ NE Z and Y ≃ NE Z.
Clearly, if X is nonevasive, then X ≃ NE pt, but is the opposite true? The answer to that is "no". To see this, consider the standard example of a space which is contractible, but not collapsible: let H be the so-called house with two rooms, see Figure 4.1.
The space H is not collapsible, hence nonevasive, see [Co68] for an argument. On the other hand, we leave it to the reader to see that it is possible to triangulate the filled cylinder C given by the equations |z| ≤ 1, x 2 + y 2 ≤ 1, so that C ց NE H.
The analogous equivalence relation, where ց NE , and ր NE , are replaced by ց, and ր, is called the simple homotopy equivalence; its equivalence classes are called simple homotopy types. The celebrated theorem of J.H.C. Whitehead states that the simplicial complexes with the simple homotopy type of a point are precisely those, which are contractible, see [Co68]. Therefore, the class of the simplicial complexes which are NE-equivalent to a point relates to nonevasiveness in the same way as contractibility refers to collapsibility. Clearly, this means that this class should constitute an interesting object of study. We conjecture that the NE-equivalence is much finer than the Whitehead's simple homotopy type. We make two conjectures: a weak and a strong one.
Conjecture 4.2. There exist finite simplicial complexes X and Y having the same simple homotopy type, such that X ≃ NE Y .
, which all have the same simple homotopy type, such that X i ≃ NE X j , for all i = j.
Again, in the simple homotopy setting, the phenomenon of the Conjectures 4.2 and 4.3 is governed by an algebraic invariant called the Whitehead torsion, namely: a homotopy equivalence between finite connected CWcomplexes is simple if and only if its Whitehead torsion is trivial, see [Co68,(22.2)]. It is enticing to hope for an existence of some similar invariant in our NE-setting.
Finally, let us remark, that whenever we have simplicial complexes X ≃ NE Y , there exists a simplicial complex Z, such that X ր NE Z ց NE Y . Indeed, assume A ց NE B ր NE C, for some simplicial complexes A, B, and C. Let S = V (A) \ V (B), and T = V (C) \ V (B). Let D be the simplicial complex obtained by attaching to A the vertices from T in the same way as they would be attached to B ⊆ A. Clearly, since the links of the vertices from S did not change, they can still be removed in the same fashion as before, and therefore we have A ր NE D ց NE C. Repeating this operation several times, and using the fact that the reductions ր NE (as well as ց NE ) compose, we prove the claim.