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Strong homotopy types, nerves and collapses

Jonathan Ariel Barmak, Elias Gabriel Minian

arXiv:0907.2954v1math.GTmath.ATmath.CO

TL;DR

Determining simplicial collapsibility is difficult, and classical cores can be nonunique. The paper introduces strong collapses and connects them to nerves and finite topological spaces, proving results for vertex-homogeneous complexes and barycentric subdivisions. Among these, strong collapsibility with vertex homogeneity forces a simplex, and collapsibility yields non-evasiveness of the barycentric subdivision.

  • Problem

    It can be difficult to determine whether a simplicial complex is collapsible, and classical cores are not unique.

    Source: §1 · p. 1

  • Method

    The paper develops strong homotopy types using strong collapses, relates cores to iterated nerves, and uses finite topological spaces to study collapse notions.

    Source: §1 · pp. 1–2

  • Results

    Strong collapsibility of a vertex-homogeneous complex forces it to be a simplex, while a collapsible K has a 1-collapsible, non-evasive barycentric subdivision K′.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    Unique cores obtainable by iterating nerves provide a distinct viewpoint for studying simplicial collapsibility.

    Source: §1 · p. 2

    The converse of the result that collapsible K has non-evasive K′ remains unknown, though it holds when K′ is strong collapsible.

    Source: §1 · p. 3, §5 · p. 18

Abstract

from arXiv · show

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of view. We analyze vertex-transitive simplicial $G$-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of finite topological spaces.

1. Introduction

The paper introduces strong homotopy types as a more manageable framework for studying simplicial collapsibility, addressing difficulties that include nonunique classical cores. It connects strong collapses to nerves and finite topological spaces and derives results on vertex-homogeneous complexes and barycentric subdivisions.

1. Introduction

  • Classical collapsibility is difficult to determine, and collapsible complexes can collapse to nontrivial subcomplexes with no free faces.The paper identifies nonuniqueness of cores as a difficulty in collapsibility and simple homotopy theory.Source: §1 · p. 1
  • A strong collapse removes the open star of a dominated vertex, whose link is a simplicial cone, and is a particular kind of classical collapse.Strong homotopy equivalence also arises from contiguity classes.Source: §1 · p. 1
  • Strong cores are unique up to isomorphism and can be reached by iterating the nerve operation.A minimal complex has no dominated vertices; a core is a minimal subcomplex reached from K by strong collapse.Source: §1 · p. 2
  • Non-evasive complexes are classically collapsible, but the converse fails; the Evasiveness conjecture asks whether a vertex-transitive non-evasive complex must be a simplex.The conjecture remains open for non-evasive complexes, while the corresponding claim is false for collapsible complexes in general.Source: §1 · p. 2
  • For vertex-homogeneous complexes, strong collapsibility implies the complex is a simplex, and the Evasiveness conjecture reduces to minimal non-evasive complexes.The square nerve is the core of any vertex-homogeneous complex; the core of a vertex-homogeneous non-evasive complex remains vertex-homogeneous and non-evasive.Source: §1 · p. 2
  • Finite topological spaces translate simplicial complexes to posets of simplices and translate non-evasive collapse exactly between the two settings.A collapse of finite spaces induces a collapse of their associated simplicial complexes, and simplicial collapse induces a collapse between associated finite spaces.Source: §1 · p. 2
  • A simplicial collapse K ց L induces a 1-collapse K′ ց1 L′; consequently, if K is collapsible, K′ is 1-collapsible and non-evasive.The converse is unknown, but holds when K′ is strong collapsible; the paper also characterizes strong collapsibility of K via K′ and of the join KL via K or L.Source: §1 · p. 3

2. Strong homotopy types

Strong homotopy types are described by elementary strong collapses, which remove dominated vertices. Their cores are unique up to isomorphism and distinguish strong homotopy types.

2. Strong homotopy types

  • A strong collapse deletes a dominated vertex whose link is a simplicial cone; a strong equivalence uses maps inverse up to contiguity.The paper also characterizes domination by requiring every maximal simplex containing the vertex to contain its dominator.Source: §2 · pp. 3–4
  • Strong collapse is stronger than ordinary simplicial collapse, and strong equivalences are simple homotopy equivalences.The paper proves both relationships.Source: §2 · pp. 4, 6
  • Every complex has a core, unique up to isomorphism; two complexes have the same strong homotopy type exactly when their cores are isomorphic.A core is a minimal subcomplex reached by strong collapses.Source: §2 · p. 5
  • A collapsible, non-evasive complex can still be minimal and therefore fail to have the strong homotopy type of a point.The homogeneous 2-complex in Figure 2 is the stated example.Source: §2 · p. 5, Figure 2 · p. 5
  • A complex is strong collapsible exactly when its barycentric subdivision is strong collapsible, although the two need not share a strong homotopy type.The boundary of a 2-simplex and its barycentric subdivision are minimal, non-isomorphic complexes.Source: §2 · p. 6
  • A join is strong collapsible exactly when at least one of its factors is strong collapsible.The paper contrasts this with the unresolved question of whether a collapsible join must have a collapsible factor.Source: §2 · p. 6

3. The nerve of a complex

The nerve preserves homotopy type, and iterating it reveals a complex’s core. In particular, the second nerve is a strong-collapse target, while reaching a point characterizes strong collapsibility.

3. The nerve of a complex

  • The nerve has maximal simplices of the original complex as vertices, and it preserves homotopy type.The homotopy equivalence is stated via Dowker’s theorem.Source: §3 · pp. 6–7
  • Every complex strongly collapses to a subcomplex isomorphic to its second nerve.The construction embeds a copy of N²(K) as a full subcomplex of K and removes the other vertices by strong collapses.Source: §3 · pp. 7–8
  • A complex is minimal exactly when it is isomorphic to its second nerve.Source: §3 · p. 8
  • Some iterated nerve of any complex is isomorphic to its core.The even nerve sequence decreases up to isomorphism until it stabilizes at a minimal complex.Source: §3 · p. 9
  • Some nerve iterate is a point exactly for strongly collapsible complexes; a contractible example instead returns to an isomorphic N²(K) and never reaches a point.Source: §3 · pp. 7, 9

4. Finite topological spaces

The section relates strong homotopy types of simplicial complexes to homotopy types of finite spaces. This correspondence yields results on cores, collapses, and barycentric subdivision.

4. Finite topological spaces

  • Removing a beat point is an elementary strong collapse and a strong deformation retraction; strong expansions of finite spaces coincide with strong deformation retracts.A beat point is one whose strict lower set has a maximum or whose strict upper set has a minimum.Source: §4 · pp. 10–11
  • Every finite space has a core, unique up to homeomorphism, and two finite spaces are homotopy equivalent exactly when their cores are homeomorphic.A core is a minimal finite space obtained by strong collapse.Source: §4 · pp. 10–11
  • For finite T0-spaces, a subset is reached by strong collapse exactly when it is a strong deformation retract.The section also shows that a core is precisely a minimal finite space obtainable by strong collapse.Source: §4 · p. 11
  • Homotopies between finite-space maps translate into contiguity of associated simplicial maps, while contiguous simplicial maps induce homotopic maps of finite spaces.These map-level correspondences underpin the relationship between finite-space homotopy and strong homotopy of complexes.Source: §4 · p. 12
  • Homotopy-equivalent finite T0-spaces correspond to associated complexes with the same strong homotopy type.Conversely, complexes with the same strong homotopy type have homotopy-equivalent associated finite spaces.Source: §4 · pp. 12, 14
  • A complex is strong collapsible if and only if its barycentric subdivision is strong collapsible.The section also proves that a finite T0-space is contractible if and only if its barycentric subdivision is contractible.Source: §4 · pp. 13–14

5. From strong collapses to evasiveness

The section uses a hierarchy of collapses to compare simplicial collapsibility with non-evasiveness, translating these notions through finite topological spaces. It shows that collapses become 1-collapses after barycentric subdivision, while examples distinguish the collapse notions and the converse to one subdivision result remains unknown.

From strong collapses to evasiveness

  • An (n + 1)-collapse deletes vertices with n-collapsible links; n-collapsibility implies (n + 1)-collapsibility, and the notions coincide with non-evasiveness through dimension n + 1.The implication is strict at every n.Source: §5 · pp. 15–16
  • Removing the maximal simplex {a, b, c, d} yields K, which is not 1-collapsible but is 2-collapsible; links of a, b, c, and d are isomorphic to the complex of Example 2.13.Figure 6 depicts a 2-collapsible simplicial complex that is not 1-collapsible.Source: §5 · p. 16, Figure 6 · p. 16
  • A collapsible complex need not be non-evasive: Figure 7’s example has a free face {1, 3}, but its vertex links are graphs that are not trees.The example is therefore evasive despite being collapsible.Source: §5 · p. 16, Figure 7 · p. 17
  • Finite-space collapses are NE-collapses, and X collapses to Y exactly when K(X) 1-collapses to K(Y); NE-collapse also transfers between the two settings.For complexes, an NE-collapse is a collapse, whereas for finite spaces a collapse is an NE-collapse.Source: §5 · pp. 17–18
  • If K collapses to L, then K′ 1-collapses to L′; in particular, a compact polyhedron has a collapsible triangulation exactly when it has a 1-collapsible triangulation.This strengthens Welker’s result that the barycentric subdivision of a collapsible complex is non-evasive.Source: §5 · p. 18
  • The converse of the result that collapsibility of K implies non-evasiveness of K′ is unknown.If K′ is strong collapsible, however, K is strong collapsible and hence collapsible.Source: §5 · p. 18

6. Vertex-homogeneous complexes

The section studies the Evasiveness conjecture for vertex-homogeneous complexes, proving it for strong collapsible complexes and reducing the general conjecture to minimal complexes. It also identifies their cores with square nerves and derives further collapsibility results.

6. Vertex-homogeneous complexes

  • Every simplicial action on a strong collapsible complex has a fixed point.The section derives this using Stong’s fixed-point theorem for contractible finite T0-spaces.Source: §6 · p. 19
  • Every strong collapsible vertex-homogeneous complex is a simplex, proving the Evasiveness conjecture for this class.The conjecture asks whether every non-evasive vertex-homogeneous complex is a simplex; its general case remains open.Source: §6 · p. 19
  • For any vertex-homogeneous complex, its core is vertex-homogeneous and isomorphic to its square nerve N 2(K).The square nerve is also isomorphic to a full subcomplex selecting one vertex from each equivalence class of ≺.Source: §6 · p. 20
  • Every vertex-homogeneous complex is isomorphic to an n-th multiple of its core; with a prime number of vertices, it is minimal or a simplex.Source: §6 · p. 21
  • For a complex K and vertex multiplicities F, n-collapsibility of F K implies n-collapsibility of K, so non-evasiveness also descends from F K to K.Source: §6 · p. 21
  • Outside the vertex-homogeneous setting, a non-evasive complex can have an evasive core.The example’s core is evasive because it is two-dimensional and not 1-collapsible.Source: §6 · p. 22
  • The Evasiveness conjecture reduces to minimal complexes because the core of a vertex-homogeneous non-evasive complex is non-evasive.The resulting equivalent conjecture is that a minimal, vertex-homogeneous, non-evasive complex is a point.Source: §6 · p. 22
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