The minimal model of a simplicial complex: algorithm and implementation

C. Alquezar-Baeta, M. Marco-Buzunáriz, J. Martin-Morales

Castellón de la Plana, June 2022

Motivation

Given a topological space X, we can define a (horribly infinitely generated) CDGA A_{PL}(X).

There exists a (hopefully) less horrible minimal model M of this algebra.

This minimal model describes the rational type of X (i.e. its homotopy up to torsion).

We have an algorithm for computing the minimal model of a CDGA of finite type.

The goal is to substitute A_{PL}(X) by a less horrible CDGA in the case of finite simplicial complexes/sets; and to adapt the algorithm for these not so horrible ones.

CDGA’s

Definition

An algebra A is graded if:

  • A=\cdots \bigoplus A^0 \bigoplus A^1\bigoplus A^2\bigoplus \cdots
  • A^iA^j\subseteq A^{i+j}
  • An element x\in A is said to be homogenous of degree i if x\in A^i.

Note that not every element is homogeneous, but they are always sum of their homogeneous parts.

Definition

A graded algebra A is said to be (graded) commutative if

\forall x\in A^i, y\in A^j, xy = (-1)^{|x||y|}yx

CDGA’s

Definition

Given a graded commutative algebra A, a differential is a linear map d:A\to A that satisfies:

  1. d(d(x)) = 0 \ \ \ \ \forall x\in A
  2. \forall x \in A^i\ \ \ \ d(x)\in A^{i+1}
  3. For every homogeneous x\in A, \ \ \ d(xy)=d(x)y+(-1)^{|x|}xd(y)

Definition

A CDGA is a commutative graded algebra, endowed with a differential

Note that the differential map d gives a cochain complex

\cdots \longrightarrow A^{n-1} \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \cdots

and hence, a cohomology H^*(A).

Examples

  • Polynomial rings
  • Exterior algebras
  • The De-Rham algebra of a smooth manifold.
  • The algebra A_{PL_n}:
    • n generators of degree 0: t_1,\ldots,t_n
    • n generators of degree 1: y_1,\ldots,y_n
    • the differential map given by d(t_i)=y_i

The algebra A_{PL_n} is an algebraic model for the DeRham algebra of an n-dimensional euclidean space.

The algebra A_{PL}

If we see the standard simplex \Delta_n\subseteq \mathbb{R}^{n+1}, we can model the differential forms in its interior with A_{PL_n} by restriction.

we can use baricentric coordinates, adding:

  • t_0= 1-t_1-\cdots -t_n
  • y_0 = -y_1-\cdots -y_n

The restriction to the faces induce the face maps

\begin{array}{ccc} \partial_i: A_{PL_n} & \longrightarrow & A_{PL_{n-1}} \\ t_0 & \mapsto & t_0 \\ & \vdots & \\ t_{i-1} & \mapsto & t_{i-1} \\ t_i & \mapsto & 0 \\ t_{i+1} & \mapsto & t_i \\ & \vdots & \\ t_n & \mapsto & t_{n-1} \end{array}

The algebra A_{PL}

The algebra A_{PL}

The algebra A_{PL}

The algebra A_{PL}(K)

Let K be a simplicial complex of dimension d, considered as a set of simplicies.

Consider the set of maps \Phi: K \to \bigoplus_{n=0}^d A_{PL_n} satisfying:

  • For each n-dimensional simplex \sigma, \Phi(\sigma)\in A_{PL_n}
  • If \sigma is the i-th face of \sigma', \Phi(\sigma)=\partial_i\Phi(\sigma')

The induced structure in the A_{PL_n} algebras endow this with a CDGA structure.

Definition:

The above algebra is called the A_{PL} algebra of the simplicial complex K

Properties of A_{PL}(K)

  • It has infinite dimension in each degree
  • H^*(A_{PL}(K),\mathbb{Q}) = H^*(|K|,\mathbb{Q})
  • We have a linear map \oint : A_{PL}(K) \to C^*(K)
    • Moreover, it is a morphism of cochain complexes
  • The minimal model gives the rational type of |K| (under reasonable conditions).
    • In particular, it allows to compute \pi_i(|K|)\oplus \mathbb{Q}.

Minimal models

Definition

A CDGA M is said to be a Sullivan algebra if:

  • M is freely generated (as a graded commutative algebra) by a well ordered set \{g_1,g_2,\ldots\}
  • The differential of each generator d(g_i) lives in the subalgebra generated by \{g_j\mid g_j < g_i\}

Definition

A Sullivan algebra is said to be minimal if the order in the set of generators is non decreasing with the degree: g_j < g_i \Longrightarrow deg(g_j)\leq deg(g_i)

Minimal models

Definition:

A minimal model of a CDGA A is a quasi-isomorphism \phi:M\longrightarrow A from a minimal Sullivan algebra M to A

Theorem

Each CDGA A with H^0(A)=\mathbb{Q} has a minimal model, and it is unique up to isomorphism.

Computation of the minimal model

We need to give:

  • A set of generators
  • Their degree
  • Their differential
  • Their image by the morphism \phi

Computation of the minimal model

Algorithm to compute the minimal model of a CDGA of finite type.

\begin{CD} \dots @>d>> M^{i} @>d>> M^{i+1} @>d>> \dots\\ @. @VV{\phi}V @VV{\phi}V @.\\ \dots @>d>> A^{i} @>d>> A^{i+1} @>d>> \dots \end{CD}

  1. Find the first i>0 such that H^i(A)\neq 0.
  2. Take a^i_1,\ldots,a_{n_i}^i such that H^i(A)=\phi^*(H^i(M))\oplus\langle [a^i_1],\ldots,[a_{n_i}^i]\rangle.
  3. Add generators x_1^i,\ldots,x_{n_i}^i with d(x_s^i)=0 and \phi(x_s^i)=a_s^i.
  4. Compute K=\langle [p^{i+1}_1],\ldots,[p^{i+1}_{m_{i+1}}]\rangle the kernel of H^{i+1}(M)\to H^{i+1}(A).
  5. If K\neq 0, add generators y_1^i,\ldots , y_{m_{i+1}}^{i} such that d(y_{s}^{i})=p^{i+1}_{s}, and \phi(y^i_s)=a_s^i with d(a_s^i)=\phi(p^{i+1}_s). Repeat 4.
  6. Increase i and go to 2.

Computation of the minimal model

Note that they involve computations of kernels, preimages, and representatives of clases on vector spaces.

In the finite dimensional case, that is usual linear algebra.

But now A^i is infinite dimensional!

Idea: use that we have the quasi-isomorphism

\begin{CD} \dots @>d>> A_{PL}^{i}(K) @>d>> A_{PL}^{i+1}(K) @>d>> \dots\\ @. @VV{\oint}V @VV{\oint}V @.\\ \dots @>d>> C^{i}(K) @>d>> C^{i+1}(K) @>d>> \dots \end{CD}

To use it, we need:

  • A way to lift cochains from C^*(K) to A_{PL}(K)
  • Given a\in A_{PL}^{i+1}(K), find some b \in A_{PL}^{i}(K) such that d(b)=a (when we can ensure that it exists).

Propagation

Consider a simplicial cochain c that assigns 1 to a n-simplex \sigma, and 0 to the rest. (That is, an element of the standard basis of C^n(K)).

We want to find an element \Phi of A_{PL}(K) such that \oint \Phi= c. That is an assignment of a polynomial to each simplex, that is compatible with the faces.

We can assign n!y_1\ldots y_n = n!dt_1\dots dt_n to \sigma, and 0 to the rest of the n-skeleton. That is compatible with the faces from that dimension down.

But for higher dimensional simplices, we have to propagate the choice in a compatible way.

Theorem:

Given any simplicial cochain \tau \in C^*(K), there exist a compatible propagation \Phi\in A_{PL}(K) such that \oint \Phi=\tau.

Propagation

  • \tau((0,1)) = 0
  • \tau((0,2)) = 0
  • \tau((0,3)) = 0
  • \tau((1,2)) = 1
  • \tau((1,3)) = 0
  • \tau((2,3)) = 0

Propagation

  • \tau((0,1)) = 0
  • \tau((0,2)) = 0
  • \tau((0,3)) = 0
  • \tau((1,2)) = 1
  • \tau((1,3)) = 0
  • \tau((2,3)) = 0
  • \Phi((0,1)) = 0
  • \Phi((0,2)) = 0
  • \Phi((0,3)) = 0
  • \Phi((1,2)) = y_1
  • \Phi((1,3)) = 0
  • \Phi((2,3)) = 0
  • \Phi((0,1,2)) = t_1y_2 - t_2y_1
  • \Phi((0,1,3)) = 0
  • \Phi((0,2,3)) = 0
  • \Phi((1,2,3)) = t_0y_1-t_1y_0
  • \Phi((0,1,2,3)) = t_1y_2-t_2y_1

Finding preimages by the differential

Let a\in A_{PL}^{i+1}(K) be a polynomial form, such that there exist some b \in A_{PL}^{i}(K) with d(b)=a.

In principle, we should search for b in an infinite dimensional space.

However, we can ensure that it lives inside a specific finite dimensional subspace (although it can be big).

So we can use standard linear algebra to find it.

Summary

  • The algorithm from Manero, M. does not work for A_{PL}(K) as-is.
  • But the steps where it doesn’t, can be circumvented using:
    • The quasi-isomorphism \oint : A_{PL}(K)\to C^*(K).
    • The lifting through propagation.
    • Restriction to suitable finite dimensional spaces.
  • So we have an algorithm to compute the minimal model of finite simplicial complexes.
  • Can be easily adapted for simplicial sets of finite type.
  • In particular, it can give us the rational homotopy type and some criteria for formality (if the space is “good”)

Thank you