Discrete Morse functions on infinite complexes...Discrete Morse functions on inﬁnite complexes...

Discrete Morse functions on infinitecomplexes

Neža Mramor Kostajoint work with

Rafael Ayala, Gregor Jerše, José Antonio Vilches

University of Ljubljana, Slovenia

Discrete, Computational and Algebraic Topology,Copenhagen 2014

Discrete Morse functions

M regular locally finite cell complex (simplicial, cubical, . . . )

A discrete Morse function F on M is a labelling of the cells withlabels t ∈ R that is monotone increasing with respect todimension, with at most one exception on each cell.

That is, for each cell σ(k) one of the following is trueI F (τ) ≥ F (σ) for exactly one face τ (k−1) < σ

I F (σ) ≥ F (ν) for exactly one coface ν(k+1) > σ

I F (τ) < F (σ) < F (ν) for all faces τ and cofaces ν.

There is an alternative nice definition due to Benedetti . . .

The gradient vector field

A discrete Morse function F on M induces a partial pairing Von the cells of M.

It consisting of pairs of cells such that

V = {(τ (k−1), σ(k)) | τ < σ,F (τ) ≥ F (σ)}.

We call V the discrete gradient vector field of F and denote itby arrows pointing from a face to its paired coface.

Cells that appear in V are reguar and cells that do not arecritical.

V = {(τ (k−1), σ(k)) | τ < σ,F (τ) ≥ F (σ)}.

Example 1

Here is an example of a discrete Morse function and theinduced gradient vector field on a torus:

A discrete Morse function on a finite cell complex has at leastone critical vertex, at the minimal value.

Example 1

Here is an example of a discrete Morse function and theinduced gradient vector field on a torus:

A discrete Morse function on a finite cell complex has at leastone critical vertex, at the minimal value.

Example 2

Here is an example on an infninite strip:

On an infinite complex a discrete Morse function can have nocritical cells.

Example 2

Here is an example on an infninite strip:

On an infinite complex a discrete Morse function can have nocritical cells.

Gradient paths

A sequences of adjoining arrows forms a gradient path or aV-path.

This is a discrete vector field on a genus 2 surface with manyV -paths.

Gradient paths

A sequences of adjoining arrows forms a gradient path or aV-path.

This is a discrete vector field on a genus 2 surface with manyV -paths.

Discrete vector fields

A discrete vector field on M is a partial pairing on the cells of M

W = {(τ (k−1), σ(k)) | τ < σ}

with every cell of M in at most one pair of W .

Discrete vector fields

A discrete vector field on M is a partial pairing on the cells of M

W = {(τ (k−1), σ(k)) | τ < σ}

with every cell of M in at most one pair of W .

Integrability

A discrete vector field on a finite complex is the gradient field ofa discrete Morse function if and only if it is acyclic (Forman).

Algorithm for assigning values:I on critical cells, the value is the dimensionI along V -paths of dimension k assign decreasing values

from k towards k − 1, in case of a comflict (where V -pathsmerge) the lowest value wins.

This gives a selfindexing discrete Morse function.

It works also for infinite locally finite complexes.

Integrability

Morse inequalities on finite complexes

The following result is well known:

Theorem (Forman)If F is a discrete Morse function on M with ck critical cells ofdimension k and bk is the k-th Betti number of M,k = 0,1, . . . ,n (n is the dimension of M). Then:

1. ck ≥ bk for all k,2. ck − ck−1 + · · · ± c0 ≥ bk − bk−1 + · · · ± b0, for all k,3. c0 − c1 + · · ·+ (−1)ncn = b0 − b1 + · · ·+ (−1)nbn = χ(M).

Even more: the discrete gradient vector field gives an algorithmfor computing the homology of M.

Example 2 above shows that Mores inequalities do not hold ingeneral on infinite complexes.

V a discrete vector field.

A k -ray in V is an infinite sequence

τ(k−1)0 < σ

(k)0 > τ

(k−1)1 < σ

(k)1 , . . .

with τi 6= τi+1 for all i = 0,1, . . ..

Two rays are equivalent if they coincide from some commoncell on.

V a discrete vector field.

A k -ray in V is an infinite sequence

τ(k−1)0 < σ

(k)0 > τ

(k−1)1 < σ

(k)1 , . . .

with τi 6= τi+1 for all i = 0,1, . . ..

Two rays are equivalent if they coincide from some commoncell on.

Descending and ascending rays

A k-ray is

I descending if the arrows go from τi to σi for all i and

I ascending, if the arrows go from τi , σi−1 for all i i .

A descending ray is an infinite gradient path.

A k-ray is

I descending if the arrows go from τi to σi for all i andI ascending, if the arrows go from τi , σi−1 for all i i .

A k-ray is

Critical elements

Theorem (Ayala, Fernándes, Vilches)If F is a discrete Morse function on an infinite locally finiteregular cell complex M with ck critical cells and dk equivalenceclasses of descending rays, and if bk is the k-th Betti number ofM, k = 0,1, . . . ,n (n is the dimension of M). Then:

1. (ck+dk )−(ck−1+dk−1)+· · ·±(c0+d0) ≥ bk−bk−1+· · ·±b0for all k = 0,1, . . . ,n − 1,

2. ck + dk ≥ bk for all k = 0,1, . . . ,n,3. (c0 + d0)− (c1 + d1) + · · ·+ (−1)ncn =

b0 − b1 + · · ·+ (−1)nbn = χ(M).

So, define the critical elements of a discrete Morse function onan infinite locally finite regular cell complex to be the criticalcells and equivalence classes of maximal descending rays.

ci and di are all finite in the above theorem.

Critical elements

b0 − b1 + · · ·+ (−1)nbn = χ(M).

Critical elements

b0 − b1 + · · ·+ (−1)nbn = χ(M).

Reversing rays

The proof of this is based on the idea of reversing rays:

Let r be a descending k -ray, pick a cell τ k−1 in r , and reverseall arrows from there on:

The descending ray is replaced by the critical cell τ and anascending ray.

Reversing rays

Multirays

Reversing rays is permissible only when no cycles aregenerated.

A cycle is generated if arrows are reversed on a ray where abypass exists:

Kukieła (in the more general context of Morse matchings onposets) defined the concept of a multiray, which is a ray alongwhich there are infinitely many bypasses, and showed thatmultirays induces an infinite number of equivalence classes ofrays.

Multirays

Sublevel complexesThe sublevel complex of F at a is

M(a) =⋃

F (σ)≤a

⋃τ≤σ

Theorem (Forman’s Main theorem)If M is finite, then

I if F−1([a,b]) contains no critical cells of F then M(b)collapses onto M(a),

I if F−1([a,b]) contains one critical cell of dimension k thenM(b) is homotopy equivalent to M(a) with a k-cell attachedalong its boundary.

In particular, M has the homotopy type of a CW complex withone cell of dimension k for each critical cell of F of dimension k .

M(a) =⋃

F (σ)≤a

⋃τ≤σ

M(a) =⋃

F (σ)≤a

⋃τ≤σ

Example

M = M(2)

F has no critical cells, in particular no critical cell in F−1([1,2]),but M(1) is not homotopy eqivalent to M(2).

Proper discrete Morse functions

The discrete Morse function F is proper if F−1([a,b]) containsat most finitely many cells for any interval [a,b]. In this caseForman’s main theorem easily generalizes:

For a proper discrete Morse function Forman’s Main theorem isvalid.

Existence of proper integrals

Which discrete vector fields have proper integrals?

No V -loops does not suffice:

For any interval [−a,a], F−1([−a,a]) is infinite.

Incident rays

The descending region of a k-cell σ consists of all kdimensional V -paths beginning in the boundary of σ. Inaddition, we add recursively all regular pairs (τ, ν) of lowerdimension in the boundary of their union with all cofaces of τexcept ν already included.

The descending region of a ray is the union of the descendingregions of its cells.

A ray r1 of dimension d1 is incident to a ray r2 of dimensiond2 > d1 when D(r1) ∩ D(r2) contains infinitely many cells.

Classification theorem

A forbidden configuration is a descending ray with an incidentascending ray of lower dimension in the boundary of itsdescending region.

Theorem (Ayala, Jerše, M, Vilches)A discrete vector field on a locally finite infinite regular cellcomplex M with finitely many critical elements admits a properintegral if and only if it has no forbidden configurations.

On complexes of dimension 1 (graphs) no V -loops suffices.

The proof is an algorithm for constructing such an integral:1. all critical cells are given the value equal to their dimension,2. M is expressed as the union M = ∪∞i=0Ki of an increasing

sequence of finite sub complexes with the property, thateach Ki intersects any ray in only one component, and thatit includes the whole closed descending region of a cellthat does not belong to any ray,

3. F is defined on K0 essentially by carefully assigningdecreasing values along V -paths,

4. F is inductively extended from Ki to Ki+1 in the same way.

Homology

Theorem (Kukieła)If M is a regular cell complex (not necessarily locally finite) withan acyclic discrete gradient field V that has a finite number ofequivalence classes of maximal descending rays then thereexists an acyclic gradient field V ′ with no descending rays, withone critical cell of dimension k for each critical cell of V of thesame dimension and in addition a critical cell of dimension k foreach descending k-ray of V .

This discrete vector field gives a Morse chain complex withhomology isomorphic to the homology of K .

Actually the theorem is proved in the more general setting ofmatchings on posets.

Homology

At the very end: an attempt at motivation

There are many interesting topological objects which can berepresented as infinite regular cell complexes.

My example: the grope, the classifying space of the gropegroup π

π is a countably generated locally free group which is not free.

It follows from the Stallings-Swan theorem its cohomologicaldimension is 2, so there exists a nontrivial 2-cocyclea ∈ H2(π;M) for some π-module M.

Dranishnikov and Rudyak: the cup product b ∪ b ∈ H2(π; I ⊗ I)of 2 copies of the Berstein class b is nontrivial.

The Berstein class b is an element of H1(G; I), whereI = kerZ[G]→ Z is the augmentation ideal.

The grope (by Aleš Vavpetic)X = lim

→(X0

ι0−→ X1ι1−→ X2

ι2−→ X3 · · · )

→(X0

ι0−→ X1ι1−→ X2

ι2−→ X3 · · · )

→(X0

ι0−→ X1ι1−→ X2

ι2−→ X3 · · · )

→(X0

ι0−→ X1ι1−→ X2

ι2−→ X3 · · · )

→(X0

ι0−→ X1ι1−→ X2

ι2−→ X3 · · · )

→(X0

ι0−→ X1ι1−→ X2

ι2−→ X3 · · · )

A discrete vector field on the grope

The homology groups H1 and H2 are trivial but . . .

a discrete vector field on the grope has infinitely manydescending rays.

Could we obtain a nontrivial dimension two cohomologyelement from (some version of) discrete Morse theory?

Thank you!

Discrete Morse functions on infinite complexes...Discrete Morse functions on inﬁnite complexes...

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