1.9 KiB
Congreso Bienal de la Real Sociedad Matemática Española 2022
Title
Effective computation of the Sullivan model of a topological space and its rational homotopy type
Abstract
The rational homotopy type of a 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 [1,2], 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 talk, we will: (1) introduce the notion of the Sullivan model of a space, (2) explain in which cases this model captures the topological information we are interested in, and (3) we will present an effective algorithm (with a concrete implementation) for the computation of the rational homotopy type of this kind of topological spaces. Finally, we will discuss the possible applications of this theory in the field of singularity theory.
References
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[1] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude, Rational Homotopy Theory, Springer Nature (2001)
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[2] Griffiths, P. and Morgan J., Rational homotopy theory and differential forms, Progress in Mathematics vol. 16, Birkhäuser (2013).
Funding
This work was partially supported by PID2020-114750GB-C31, E22_20R: Algebra Y Geometría and BES-2017-079979.