Simplicial Complexes and Δ-Complexes

This note expands some of the material on Δ-complexes in §2.1 of Hatcher’s book Algebraic Topology, from a slightly more classical point of view.

Simplices Given any linearly independent set α = {v_{0}, v_{1}, …, v_{n}} of n + 1 points in ℝ^{q}, the n-simplex with vertices α is the convex hull of α, i. e. the set of all points of the form (see pp. 102–3)

t_{0}v_{0} + t_{1}v_{1} + … + t_{n}v_{n},  where ∑_{i=0}^{n} t_{i} = 1 and t_{i} ≥ 0 for all i.  (1)

Let us call it Δ_{α}. (One can relax the condition on α slightly; we only need the n vectors v_{1} − v_{0}, …, v_{n} − v_{0} to be linearly independent.)

Low-dimensional examples are familiar enough: for n = 1 we find the line segment joining v_{0} to v_{1}; for n = 2 we find the triangle with vertices v_{0}, v_{1} and v_{2}; for n = 3 we find the tetrahedron with vertices α. The definition makes sense even for n = 0; we simply obtain the one-point space {v_{0}} as the 0-simplex with one vertex v_{0}. However, the empty set is not considered a simplex.

Definition 2 The standard n-simplex Δ^{n} ⊂ ℝ^{n+1} has the standard basis of ℝ^{n+1}, {e_{0}, e_{1}, …, e_{n}}, as its set of vertices, i. e.

Δ^{n} = {(t_{0}, t_{1}, …, t_{n}) ∈ ℝ^{n+1} : ∑_{i=0}^{n} t_{i} = 1, t_{i} ≥ 0 for all i} .  (3)

Given 0 ≤ i ≤ n, the i-th face F_{i} of Δ^{n} is the subspace of points that satisfy t_{i} = 0; it is the (n−1)-simplex whose vertices are all those of Δ^{n} except e_{i}. The geometric boundary ∂Δ^{n} of Δ^{n} is the subspace ⋃_{i=0}^{n} F_{i}. (Some books adopt a different choice of standard n-simplex Δ^{n}, as having vertices {0, e_{1}, e_{2}, …, e_{n}} in ℝ^{n}, but we see below that it does not really matter.)

For each i, we have the obvious linear map η_{i}: Δ^{n−1} ≅ F_{i} ⊂ Δ^{n}, defined by

η_{i}(e_{j}) = \{ \begin{matrix}e_{j} & if j < i; \\ e_{j+1} & if j ≥ i. \end{matrix}

Given α = {v_{0}, v_{1}, …, v_{n}}, the linear map T: ℝ^{n+1} → ℝ^{q} given by

T (t_{0}, t_{1}, …, t_{n}) = t_{0}v_{0} + t_{1}v_{1} + … + t_{n}v_{n}

induces the obvious continuous bijection σ_{α}: Δ^{n} → Δ_{α} (see p. 103), which is a homeomorphism since Δ^{n} is compact and Δ_{α} is Hausdorff.

Finite simplicial complexes in ℝ^{q} Let V be a linearly independent set of points in ℝ^{q}. For each subset α ⊂ V, we have defined the simplex Δ_{α}. The subspace X of ℝ^{q} formed by taking the union of some of these simplices is called a (geometric) simplicial complex, or polyhedron. Its n-skeleton X^{n} ⊂ X is formed by keeping only the i-simplices for i ≤ n. Since there is a homeomorphism (Δ^{n}, ∂Δ^{n}) ≅ (D^{n}, S^{n−1}), it is clear that X is a finite cw-complex, with one n-cell for each n-simplex. Despite appearances, simplicial complexes include many spaces of interest. Indeed, every finite cw-complex is homotopy equivalent to a polyhedron.

It is easy to see that the space X depends only on which subsets α ⊂ V are used, not on the choice of embedding V ⊂ ℝ^{q}. (Indeed, the condition on V may be relaxed somewhat, but this does not concern us.) Clearly, if Δ_{α} ⊂ X and β ⊂ α, then Δ_{β} ⊂ Δ_{α} ⊂ X. This leads to the following combinatorial description (see p. 107). We also need to allow V to be infinite, and to free ourselves of reliance on ℝ^{q}, which may suggest an inappropriate topology on X in this generality.

Abstract simplicial complexes

Definition 4 An abstract simplicial complex K consists of a set V, whose elements are called vertices, and a collection S of finite non-empty subsets of V that satisfies the axioms:

(i) For each v ∈ V, the singleton {v} ∈ S; (ii) If α ∈ S and β ⊂ α is non-empty, then β ∈ S.

A member α ∈ S consisting of n + 1 elements is called an n-simplex of K.

As before, the n-skeleton K^{n} of K is obtained by keeping only the i-simplices for i ≤ n.

One can construct a space, the geometric realization |K| of K. For each simplex α, take a copy Δ_{α} of Δ^{n}, where n is the dimension of α. Whenever β ⊂ α, we identify Δ_{β} with a subspace of Δ_{α} in order to construct the space |K| as a quotient space of the disjoint union ∐_{α} Δ_{α}. (It is enough to make the identifications for the faces, or (n−1)-simplices, β ⊂ α.)

We do not pursue these details any further. We note that the homeomorphism σ_{α}: Δ^{n} ≅ Δ_{α} is not well defined until we choose an ordering of the set α. For future purposes, we need a preferred σ_{α}, which leads to the following concept.

Ordered simplicial complexes

Definition 5 An ordered abstract simplicial complex K is an abstract simplicial complex as above, in which the set V of vertices is ordered. (Again, it suffices and is occasionally useful to assume rather less than a total ordering on V, specifically that V is partially ordered in a way that totally orders each simplex of K.) We write K_{n} for the set of n-simplices of K. Then for n > 0 and 0 ≤ i ≤ n, we define the face operator d_{i}: K_{n} → K_{n−1} by

d_{i}({v_{0}, v_{1}, …, v_{n}}) = {v_{0}, v_{1}, …, v_{i−1}, v̂_{i}, v_{i+1}, …, v_{n}},

where the vertices are understood to be in order, v_{0} < v_{1} < … < v_{n}, and the notation v̂_{i} indicates that v_{i} is to be omitted.

Lemma 6 In an ordered abstract simplicial complex K, we have

d_{i} ^{∘} d_{j} = d_{j−1} ^{∘} d_{i}: K_{n} − ; − ; → K_{n−2} whenever n ≥ 2 and i < j.  (7)

Proof Both composites delete v_{i} and v_{j} from α (as above). This is clear for d_{i}d_{j}α. To compute d_{j−1}d_{i}α, we first delete v_{i}. This causes v_{j} to move left one place to position j − 1, where it is deleted by d_{j−1}.

Geometric realization We construct the geometric realization |K| of an ordered abstract simplicial complex in a way that generalizes the finite case in ℝ^{q}. It will have a well-defined map

σ_{α}: Δ^{n} ≅ Δ^{n} × {α} − ; − ; → |K|  (8)

for each simplex α of K, where n is the dimension of α. To ensure that the faces of σ_{α}(Δ^{n}) really are what they should be, we require for each i the commutative triangle

\xymatrix{Δ^{n} \ar[r]^{σ_{α}} & |K| \\ Δ^{n−1} \ar[u]^{η_{i}} \ar[ur]_{σ_{d_{i}α}}}  (9)

The construction is by induction on the skeletons K^{n} of K. It begins with the 0-skeleton |K^{0}| = Δ^{0} × K_{0} ≅ K_{0}, where we treat each K_{n} as a discrete space and for each α ∈ K_{0}, σ_{α}: Δ^{0} → |K^{0}| ≅ K_{0} has image α.

Suppose we have already constructed the (n−1)-skeleton |K^{n−1}|, with maps σ_{β}: Δ^{i} → |K^{n−1}| for all simplices β of any dimension i ≤ n − 1, satisfying condition (9) (when n ≥ 2). We construct |K^{n}| from it by attaching n-cells, one for each α ∈ K_{n}, by the pushout square

\xymatrix{Δ^{n} × K_{n} \ar@{-->}[r]^{Φ} & |K^{n}| \\ ∂Δ^{n} × K_{n} \ar[r]^{ϕ} \ar[u]_{∪} & |K^{n−1}| \ar@{-->}[u]_{∪}}

For convenience, we restrict attention to the α-th n-simplex Δ^{n} × α and suppress the “×α” wherever possible. Its attaching map ϕ_{α}: ∂Δ^{n} → |K^{n−1}| is essentially ϕ|(∂Δ^{n}×α). In order to satisfy condition (9), we require the restriction of ϕ_{α} to the i-th face F_{i} of Δ^{n} to be

f_{α}|F_{i}: F_{i} − ; − ; →^{≅} Δ^{n−1} − ; − ; − ; →^{σ_{d_{i}α}} |K^{n−1}|.

Then Φ|(Δ^{n}×α) will provide the desired characteristic map of the α-th n-cell,

σ_{α}: (Δ^{n}, ∂Δ^{n}) − ; − ; → (|K^{n}|, |K^{n−1}|).

We need to check that ϕ_{α} is well defined on F_{i} ∩ F_{j} whenever i < j (assuming n ≥ 2; if n = 1 there is nothing to prove). On F_{i} ∩ F_{j}, considered as a subspace of F_{i}, the map F_{i} ∩ F_{j} → |K^{n−1}| is given by the upper route in the commutative diagram

\xymatrix{F_{i} & Δ^{n−1} \ar[l]_{≅}^{η_{i}} \ar[r]^{σ_{d_{i}α}} & |K^{n−1}| \\ F_{i} ∩ F_{j} \ar[u]^{∪} & F′_{j−1} \ar[l]_{≅}^{η_{i}} \ar[u]^{∪} & Δ^{n−2} \ar[ul]_{η_{j−1}} \ar[l]_{≅} \ar[u]_{σ_{d_{j−1}d_{i}α}}}

where F′_{k} denotes the k-th face of Δ^{n−1}. We note that because i ≤ j − 1, the homeomorphism η_{i}: Δ^{n−1} ≅ F_{i} carries F′_{j−1} (not F′_{j}) to F_{i} ∩ F_{j}.

If instead we regard F_{i} ∩ F_{j} as a subspace of F_{j}, we obtain

\xymatrix{F_{j} & Δ^{n−1} \ar[l]_{≅}^{η_{j}} \ar[r]^{σ_{d_{j} α}} & |K^{n−1}| \\ F_{i} ∩ F_{j} \ar[u]^{∪} & F′_{i} \ar[l]_{≅}^{η_{j}} \ar[u]^{∪} & Δ^{n−2} \ar[ul]_{η_{i}} \ar[l]_{≅} \ar[u]_{σ_{d_{i}d_{j} α}}}

The two maps Δ^{n−2} ≅ F_{i} ∩ F_{j} ⊂ Δ^{n} both coincide with the obvious linear orderpreserving map Δ^{n−2} → Δ^{n} that omits the vertices e_{i} and e_{j}. It follows from equation (7) that ϕ_{α}|F_{i} and ϕ_{α}|F_{j} agree on F_{i} ∩ F_{j}. This completes the induction step.

To make |K| a cw-complex, we topologize it as a quotient space of ⋃_{n=0}^{∞} Δ^{n} ×K_{n}. A triangulation of any space X is a homeomorphism |K| ≅ X for some simplicial

complex K.

Δ-complexes In the above geometric realization, we observe that we nowhere used the fact that each α is a set of vertices. We therefore generalize.

Definition 10 A Δ-set, or abstract Δ-complex K, consists of a sequence of sets K_{0}, K_{1}, K_{2}, …, together with face operators d_{i}: K_{n} → K_{n−1} for n > 0 and 0 ≤ i ≤ n that satisfy equation (7).

We construct its geometric realization |K| exactly as above; the result is a Δcomplex as described on p. 104. The only fact needed was Lemma 6, which we took here as an axiom.

This does not enlarge the class of spaces |K|, as Exercise 23 shows how to triangulate |K| as a simplicial complex for any Δ-complex K. However, Δ-complexes are often convenient to work with in practice, as they typically contain many fewer simplices than any simplicial triangulation of |K|. (See the foundational examples on p. 102.)