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The singular (co)chains and rational homotopy theory, I

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Felix Wierstra
Mon, 2019-07-01 15:00 - 16:00
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar

The central theme of this talk is the question: How much of the rational homotopy type of a space can we deduce from the (co)chains on the space? More precisely, in this talk we explain what the relationship is between the singular (co)chains and various approaches to the rational homotopy theory. 

Bousfield and Kan showed that there are two possible approaches to non-simply-connected homotopy theory. Both their approaches are given by a completion on the level of spaces. The first goal of this talk is to explain the relationship between these completions and the singular chains. On the singular chains with rational coefficients of a space there are several different notions of equivalence, the most important ones are given by quasi-isomorphisms and $\Omega$-quasi-isomorphisms, which are maps that become a quasi-isomorphisms after applying the cobar construction. In the first part of the talk we show that these two notions of weak equivalence correspond to the two completions of Bousfield and Kan.
The second goal of the talk focuses on the cochains on a simply-connected space. A famous theorem by Sullivan shows that two simply-connected spaces are rationally equivalent if and only if their commutative algebras of polynomial de Rham forms can be connected by a zig-zag of quasi-isomorphisms of 
commutative algebras. Since the polynomial de Rham forms are quasi-isomorphic to the singular cochains as associative algebras it is a natural question to ask whether the cochains seen as an associative algebra also determine the rational homotopy type of a space. We will show that this is indeed the case, by showing that the much more general statement that two commutative algebras can be connected by a zig-zag of commutive quasi-isomorphisms if and only if they can be connected by a zig-zag of associative quasi-isomorphisms.
This talk is based on a combination of two projects, one joint with Manuel Rivera and Mahmoud Zeinalian and one joint with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.


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