# The minimal Sullivan model of a simplicial complex

The rational homotopy type of a topological space is a simplified version of the homotopy type, where all homotopy groups are tensored by $\mathbb{Q}$. Despite the lost information, rational homotopy has the advantage of being constructive. Due to Sullivan, for a particular topological space $X$ that satisfies some conditions, it is known theoretically how to obtain its rational homotopy type via the construction of a commutative differential graded algebra, called the Sullivan model of $X$. This algebra is quasi-isomorphic to the normalized singular cochain algebra of $X$, $C^*(X)$, and it allows us to establish a categorical equivalence between homotopy types of spaces and isomorphism classes of Sullivan models: 

$$
\begin{Bmatrix}
Rational \:\: homotopy
\\ 
types \:\: of \:\:spaces
\end{Bmatrix}
\Leftrightarrow 
\begin{Bmatrix}
Isomorphim \:\:classes \:\: of 
\\ 
minimal \:\: Sullivan \:\: models
\end{Bmatrix}
$$

In this work it is presented an effective method to compute the Sullivan minimal model for a simply-connected topological space, and an implementation of such method in a Computer Algebra System. In order to illustrate such method, examples of some computations are included.

This project has been partially supported by PID2020-114750GB-C31 and BES-2017-079979, funded by MCIN/AEI/10.13039/501100011033 and also partially funded by the Departamento de Ciencia, Universidad y Sociedad del Conocimiento of the Gobierno de Aragón (Grupo de referencia E22 20R “Álgebra y Geometría”).