Categories, posets, Alexandrov spaces, simplicial complexes, with emphasis on finite spaces

J.P. May November 10, 2008

[figure: Spaces; S; π_{1}; Groups; K(−,1); i; Simp. Sets; Sd; |−|; _{Sd_{2}}; τ_{1}; Reg. Simp. Sets; Sd; Cats; _{Sd_{2}}; N; Simp. Complexes; i; 𝒳; P osets; ≅; i; T_{0}A − spaces; ≅; 𝒦]

Simplicial sets and subdivision

(Any new results are due to Rina Foygel) Δ ≡ standard simplicial category. Δ[n] is represented on Δ by n. It is Nn, where n is the poset {0, 1, ⋯, n}. Sd Δ[n] ≡ Δ[n]′ ≡ Nsd n, where sd n ≡ n′ ≡ monos/n. Sd K ≡ K ⊗_{Δ} Δ′. Lemma 1 Sd K ≅ Sd L does not imply K ≅ L but does imply K_{n} ≅ L_{n} as sets, with corresponding simplices having corresponding faces.

Regular simplicial complexes

A nondegenerate x ∈ K_{n} is regular if the subcomplex [x] it generates is the pushout of

\xymatrix{Δ[n] & Δ[n − 1] \ar[l]_{δ^{n}} \ar[r]^{d_{n}x} & [d_{n}x].}

K is regular if all x are so.

Theorem 1 For any K, Sd K is regular.

Theorem 2 If K is regular, then |K| is a regular CW complex: (e^{n}, ∂e^{n}) ≅ (D^{n}, S^{n−1}) for all closed n-cells e.

Theorem 3 If X is a regular CW complex, then X is triangulable; that is X is homeomorphic to some |i(K)|.

Properties of simplicial sets K

Let x ∈ K_{n} be a nondegenerate simplex of K. A: For all x, all faces of x are nondegenerate. B: For all x, x has n + 1 distinct vertices. C: Any n + 1 distinct vertices are the vertices

of at most one x. Lemma 2 K has B iff for all x and all monos

α, β: m ⟶ n, α^{∗}x = β^{∗}x implies α = β. Lemma 3 If K has B, then K has A. No other general implications among A, B, C.

Properties A, B, C and subdivision Lemma 4 K has A iff Sd K has A. Lemma 5 K has A iff Sd K has B. Lemma 6 K has B iff Sd K has C.

Characterization of simplicial complexes Lemma 7 K has A iff Sd^{2}K has C,

and then Sd^{2}K also has B. Lemma 8 K has B and C iff K ∈ Im(i). Theorem 4 K has A iff Sd^{2}K ∈ Im(i).

Subdivision and horn-filling

Lemma 9 If Sd K is a Kan complex, then

K is discrete. Lemma 10 If K does not have A, then Sd K

cannot be a quasicategory. Relationship of the properties to categories Theorem 5 If K has A, then Sd K ∈ Im(N). Proof: Check the Segal maps criterion.

Definition 1 A category 𝒞 satisfies A, B, or C if N𝒞 satisfies A, B, or C.

Lemma 11 𝒞 has A iff for any i: C ⟶ D and r: D ⟶ C such that r ∘ i = id, C = D and i = r = id. (Retracts are identities.)

Lemma 12 𝒞 has B iff for any i: C ⟶ D and r: D ⟶ C, C = D and i = r = id.

Lemma 13 𝒞 has B and C iff 𝒞 is a poset.

Definition 2 Define a category T 𝒞: Objects: nondegenerate simplices of N𝒞. e.g.

C = C_{0} ⟶ C_{1} ⟶ ⋯ ⟶ C_{q}

D = D_{0} ⟶ C_{1} ⟶ ⋯ ⟶ D_{r}

Morphisms: maps C ⟶ D are maps α: q ⟶ r in Δ such that α^{∗}D = C (implying α is mono). Quotient category sd𝒞 with the same objects:

α ∘ β_{1} ∼ α ∘ β_{2}: C ⟶ D

if σ ∘ β_{1} = σ ∘ β_{2} for a surjection σ: p ⟶ q such that α^{∗}D = σ^{∗}C (α: p ⟶ r, β_{i}: q ⟶ p).

(β_{i}^{∗}α^{∗}D = β_{i}^{∗}σ^{∗}C = C, i = 1, 2)

(Anderson, Thomason, Fritsch-Latch, del Hoyo)

Lemma 14 For any 𝒞, T 𝒞 has B. Corollary 1 For any 𝒞, sd𝒞 has B. Lemma 15 𝒞 has B iff sd𝒞 is a poset. Theorem 6 For any 𝒞, sd^{2}𝒞 is a poset. Compare with K has A iff Sd^{2}K ∈ Im(i). Del Hoyo: Equivalence ε: sd𝒞 ⟶ 𝒞. (Relate to equivalence ε: Sd K ⟶ K?)

Left adjoint τ_{1} to N (Gabriel–Zisman). Objects of τ_{1}K are the vertices. Think of 1-simplices y as maps

d_{1}y ⟶ d_{0}y,

form the free category they generate, and impose the relations

s_{0}x = id_{x} for x ∈ K_{0}

d_{1}z = d_{0}z ∘ d_{2}z for z ∈ K_{2}.

The counit ε: τ_{1}N𝒜 ⟶ 𝒜 is an isomorphism. τ_{1}K depends only on the 2-skeleton of K. When K = ∂Δ[n] for n > 2, the unit η: K ⟶ Nτ_{1}K is the inclusion ∂Δ[n] ⟶ Δ[n].

Direct combinatorial proof: Theorem 7 For any 𝒞, sd 𝒞 ≅ τ_{1} Sd N𝒞. Corollary 2 ε = τ_{1}ε: sd 𝒞 ⟶ τ_{1}N𝒞 ≅ 𝒞. Corollary 3 𝒞 has A iff Sd N𝒞 ≅ N sd𝒞. Remark 1 Even for posets P and Q, sdP ≅ sdQ does not imply P ≅ Q. In the development above, there is a counterexample to the converse of each implication that is not stated to be iff. Sheds light on Thomason model structure.

Alexandrov and finite spaces

Alexandrov space, abbreviated A-space: ANY intersection of open sets is open. Finite spaces are A-spaces. T_{0}-space: topology distinguishes points. Kolmogorov quotient K(A). McCord: A ⟶ K(A) is a homotopy equivalence. Space = T_{0}-A-space from now on T_{1} finite spaces are discrete, but any finite X has a closed point.

Define

U_{x} ≡ ∩{U|x ∈ U}

{U_{x}} is unique minimal basis for the topology.

x ≤ y ≡ x ∈ U_{y}; that is , U_{x} ⊂ U_{y} Transitive and reflexive; T_{0} = ⇒ antisymmetric. For a poset X, define U_{x} ≡ {y|x ≤ y}: basis for a T_{0}-A-space topology on the set X. f: X ⟶ Y is continuous ⟺ f preserves order. Theorem 8 The category 𝒫 of posets is isomorphic to the category 𝒜 of T_{0}-A-spaces.

Finite spaces: f: X ⟶ X is a homeomorphism iff f is one-to-one or onto.

Can describe n-point topologies by restricted kind of n × n-matrix and enumerate them.

Combinatorics: count the isomorphism classes of posets with n points; equivalently count the homeomorphism classes of spaces with n points. HARD! For n = 4, X = {a, b, c, d}, 33 topologies, with bases as follows:

1 all
2 a, b, c, (a,b), (a,c), (b,c), (a,b,c)
3 a, b, c, (a,b), (a,c), (b,c), (a,b,c), (a,b,d)
4 a, b, c, (a,b), (a,c), (b,c), (a,d), (a,b,c), (a,b,d), (a,c,d)
5 a, b, (a,b)
6 a, b, (a,b), (a,b,c)
7 a, b, (a,b), (a,c,d)
8 a, b, (a,b), (a,b,c), (a,b,d)
9 a, b, (a,b), (a,c), (a,b,c)
10 a, b, (a,b), (a,c), (a,b,c), (a,c,d)
11 a, b, (a,b), (a,c), (a,b,c), (a,b,d)
12 a, b, (a,b), (c,d), (a,c,d), (b,c,d)
13 a, b, (a,b), (a,c), (a,d), (a,b,c), (a,b,d)
14 a, b, (a,b), (a,c), (a,d), (a,b,c), (a,b,d), (a,c,d)
15 a
16 a, (a,b)
17 a, (a,b), (a,b,c)
18 a, (b,c), (a,b,c)
19 a, (a,b), (a,c,d)
20 a, (a,b), (a,b,c), (a,b,d)
21 a, (b,c), (a,b,c), (b,c,d)
22 a, (a,b), (a,c), (a,b,c)
23 a, (a,b), (a,c), (a,b,c), (a,b,d)
24 a, (c,d), (a,b), (a,c,d)
25 a, (a,b), (a,c), (a,d), (a,b,c), (a,b,d), (a,c,d)
26 a, (a,b,c)
27 a, (b,c,d)
28 (a,b)
29 (a,b), (c,d)
30 (a,b), (a,b,c)
31 (a,b), (a,b,c), (a,b,d)
32 (a,b,c)
33 none

Homotopies and homotopy equivalence f, g: X ⟶ Y: f ≤ g if f(x) ≤ g(x) ∀ x ∈ X. Proposition 1 X, Y finite. f ≤ g implies f ≃ g. Proposition 2 If y ∈ U ⊂ X with U open (or closed) implies U = X, then X is contractible. If X has a unique maximum or minimal point, X is contractible. Each U_{x} is contractible. Definition 3 Let X be finite.

(a) x ∈ X is upbeat if there is a y > x such that z > x implies z ≥ y.

(b) x ∈ X is downbeat if there is a y < x such that z < x implies z ≤ y.

Upbeat:

[figure: z_{1}; z_{2}; ⋯; z_{s}; y; x]

Downbeat: upside down. X is minimal if it has no upbeat or downbeat points. A core of X is a subspace Y that is minimal and a deformation retract of X. Stong: Theorem 9 Any finite X has a core. Theorem 10 If f ≃ id: X ⟶ X, then f = id. Corollary 4 Minimal homotopy equivalent finite spaces are homeomorphic.

REU results of Alex Fix and Stephen Patrias Can now count homotopy types with n points. Hasse diagram Gr(X) of a poset X: directed graph with vertices x ∈ X and an edge x → y if y < x but there is no other z with x ≤ z ≤ y. Translate minimality of X to a property of Gr(X) and count the number of such graphs. Find a fast enumeration algorithm. Run it on a computer. Get number of homotopy types with n points. Compare with number of homeomorphism types.

\begin{matrix}n & ≃ & ≅ \\ 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 5 \\ 4 & 5 & 16 \\ 5 & 9 & 63 \\ 6 & 20 & 318 \\ 7 & 56 & 2,045 \\ 8 & 216 & 16,999 \\ 9 & 1,170 & 183,231 \\ 10 & 9,099 & 2,567,284 \\ 11 & 101,191 & 46,749,427 \\ 12 & 1,594,293 & 1,104,891,746 \end{matrix}

Exploit known results from combinatorics.

Astonishing conclusion:

Theorem 11 (Fix and Patrias) The number of homotopy types of finite T_{0}-spaces is asymptotically equivalent to the number of homeomorphism types of finite T_{0}-spaces.

T_{0}-A-spaces and simplicial complexes Category 𝒜 of T_{0}-A-spaces (= posets); Category ℬ of simplicial complexes. McCord: Theorem 12 There is a functor 𝒦 : 𝒜 ⟶ ℬ and a natural weak equivalence

ψ: |𝒦 (X)| ⟶ X.

The n-simplices of 𝒦 (X) are

{x_{0}, ⋯, x_{n}|x_{0} < ⋯ < x_{n}},

and ψ(u) = x_{0} if u is an interior point of the simplex spanned by {x_{0}, ⋯, x_{n}}.

Let Sd K be the barycentric subdivision of a simplicial complex K; let b_{σ} be the barycenter of a simplex σ.

Theorem 13 There is a functor 𝒳 : ℬ ⟶ 𝒜 and a natural weak equivalence

ϕ: |K| ⟶ 𝒳 (K).

The points of 𝒳 (K) are the barycenters b_{σ} of

simplices of K and b_{σ} < b_{τ} if σ ⊂ τ.

𝒦 (𝒳 (K)) = Sd K and

ϕ_{K} = ψ_{𝒳 (K)} : |K| ≅ |Sd K | ⟶ 𝒳 (K).

Problem: not many maps between finite spaces!

Solution: subdivision: Sd X ≡ 𝒳 (𝒦 (X)).

Theorem 14 There is a natural weak equiv.

ξ: Sd X ⟶ X.

Classical result and an implied analogue:

Theorem 15 Let f: |K| ⟶ |L| be continuous, where K and L are simplicial complexes, K finite. For some large n, there is a simplicial map g: K^{(n)} ⟶ L such that f ≃ |g|.

Theorem 16 Let f: |𝒦 (X)| ⟶ |𝒦 (Y)| be continuous, where X and Y are T_{0}-A-spaces, X finite. For some large n there is a continuous map g: X^{(n)} ⟶ Y such that f ≃ |𝒦 (g)|.

Definition 4 Let X be a space. Define the non-Hausdorff cone ℂX by adjoining a new point + and letting the proper open subsets of ℂX be the non-empty open subsets of X.

Define the non-Hausdorff suspension 𝕊X by adjoining two points + and − such that 𝕊X is the union under X of two copies of ℂX.

Let SX be the unreduced suspension of X.

Definition 5 Define a natural map

γ = γ_{X}: SX ⟶ 𝕊X

by γ(x, t) = x if −1 < t < 1 and γ(±1) = ±.

Theorem 17 γ is a weak equivalence.

Corollary 5 𝕊^{n}S^{0} is a minimal finite space with 2n + 2 points, and it is weak equivalent to S^{n}.

The height h(X) of a poset X is the maximal length h of a chain x_{1} < ⋯ < x_{h} in X.

h(X) = dim |𝒦 (X)| + 1.

Barmak and Minian:

Proposition 3 Let X ≠ ∗ be a minimal finite space. Then X has at least 2h(X) points. It has exactly 2h(X) points if and only if it is homeomorphic to 𝕊^{h(X)−1} S^{0}.

Corollary 6 If |𝒦 (X)| is homotopy equivalent to a sphere S^{n}, then X has at least 2n + 2 points, and if it has exactly 2n + 2 points it is homeomorphic to 𝕊^{n}S^{0}.

Remark 2 If X has six elements, then h(X) is 2 or 3. There is a six point finite space that is weak homotopy equivalent to S^{1} but is not homotopy equivalent to 𝕊S^{0}.

Really finite H-spaces

Let X be a finite space and an H-space with unit e: x → ex and x → xe are each homotopic to the identity. Stong:

Theorem 18 If X is minimal, these maps are homeomorphisms and e is both a maximal and a minimal point of X, so {e} is a component.

Theorem 19 X is an H-space with unit e iff e is a deformation retract of its component in X. Therefore X is an H-space iff a component of X is contractible. If X is a connected H-space, X is contractible.

Hardie, Vermeulen, Witbooi:

Let 𝕋 = 𝕊S^{0}, 𝕋′ = Sd 𝕋.

Brute force write it down proof (8 × 8 matrix)

Example 1 There is product 𝕋′×𝕋′ ⟶ 𝕋 that realizes the product on S^{1} after realization.

Finite groups and finite spaces

X, Y finite T_{0}-spaces and G-spaces. Stong:

Theorem 20 X has an equivariant core, namely a sub G-space that is a core and a G-deformation retract of X.

Corollary 7 Let Xbe contractible. Then X is G-contractible and has a point fixed by every self-homeomorphism.

Corollary 8 If f: X ⟶ Y is a G-map and a homotopy equivalence, then it is a G-homotopy equivalence.

Quillen’s conjecture

G finite, p prime. 𝒮_{p}(G): poset of non-trivial p-subgroups of G, ordered by inclusion. G acts on 𝒮_{p}(G) by conjugation. 𝒜_{p}(G): Sub G-poset of p-tori. p-torus ≡ elementary Abelian p-group. r_{p}(G) is the rank of a maximal p-torus in G.

[figure: |𝒦 𝒜_{p}(G)|; |𝒦 (i)|; ψ; |𝒦 𝒮_{p}(G)|; ψ; 𝒜_{p}(G); i; 𝒮_{p}(G)]

Vertical maps ψ are weak equivalences.

Proposition 4 If G is a p-group, 𝒜_{p}(G) and 𝒮_{p}(G) are contractible.

Note: genuinely contractible, not just weakly.

Proposition 5 i: 𝒜_{p}(G) ⟶ 𝒮_{p}(G) is a weak equivalence.

Example 2 If G = Σ_{5}, 𝒜_{p}(G) and 𝒮_{p}(G) are not homotopy equivalent.

P ∈ 𝒮_{p}(G) is normal iff P is a G-fixed point.

Theorem 21 If 𝒮_{p}(G) or 𝒜_{p}(G) is contractible, then G has a non-trivial normal p-subgroup. Conversely, if G has a non-trivial normal p-subgroup, then 𝒮_{p}(G) is contractible, hence 𝒜_{p}(G) is weakly contractible.

Conjecture 1 (Quillen) If 𝒜_{p}(G) is weakly contractible, then G contains a non-trivial normal p-subgroup.

Easy: True if r_{P} (G) ≤ 2. Quillen: True if G is solvable. Aschbacker and Smith: True if p > 5 and G has no component U_{n}(q) with q ≡ −1 (mod p) and q odd. (Component of G: normal subgroup that is simple modulo its center). Horrors: proof from the classification theorem. Their 1993 article summarizes earlier results. And as far as Jon Alperin and I know, that is where the problem stands. Finite space version may not help with the proof, but is intriquing.