C. Alquezar-Baeta, M. Marco-Buzunáriz, J. Martin-Morales
Castellón de la Plana, June 2022
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.
An algebra A is graded if:
Note that not every element is homogeneous, but they are always sum of their homogeneous parts.
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
Given a graded commutative algebra A, a differential is a linear map d:A\to A that satisfies:
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).
The algebra A_{PL_n} is an algebraic model for the DeRham algebra of an n-dimensional euclidean space.
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:
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}
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:
The induced structure in the A_{PL_n} algebras endow this with a CDGA structure.
The above algebra is called the A_{PL} algebra of the simplicial complex K
A CDGA M is said to be a Sullivan algebra if:
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)
A minimal model of a CDGA A is a quasi-isomorphism \phi:M\longrightarrow A from a minimal Sullivan algebra M to A
Each CDGA A with H^0(A)=\mathbb{Q} has a minimal model, and it is unique up to isomorphism.
We need to give:
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}
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:
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.
Given any simplicial cochain \tau \in C^*(K), there exist a compatible propagation \Phi\in A_{PL}(K) such that \oint \Phi=\tau.
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.