String topology and homotopy Frobenius algebras

Informazioni relative al progetto

StringFrob

ID dell’accordo di sovvenzione: 896370

DOI

10.3030/896370

Progetto concluso il 31 Dicembre 2022

Data della firma CE 9 Aprile 2020

Data di avvio 1 Maggio 2021

Data di completamento 28 Dicembre 2023

Finanziato da

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Costo totale

€ 207 312,00

Contributo UE

€ 207 312,00

Coordinato da

KOBENHAVNS UNIVERSITET
Denmark

Periodic Reporting for period 1 - StringFrob (String topology and homotopy Frobenius algebras)

Periodo di rendicontazione: 2021-05-01 al 2023-04-30

Sintesi del contesto e degli obiettivi generali del progetto

String topology is the study of loops in a space. More precisely, there are interesting operations arising from the fact that whenever we have a collection of loops that intersect, we can cut and re-glue the corresponding loops. It turns out that such operations also naturally arise when studying certain quantum field theories. A natural question one can ask is: "How much do those loops together with the cutting-and-gluing operations know about the underlying space?". More precisely, given two spaces and a way to relate loops on them, what are conditions such that the corresponding cutting-and-gluing operations are the same? If the spaces are homotopy-equivalent (that is the shape of one can be deformed into the other) then we can relate loops on them and we can ask the previous question. The hope is, that knowing those extra operations will give a stronger condition on two spaces being equivalent. The main result of this action is that this is indeed the case. More precisely, it was shown that if we consider loop intersecting themselves, then the two spaces have to be simple-homotopy equivalent (under suitable conditions). Simple-homotopy equivalence is in particular a stronger condition than homotopy equivalence and measures if it possible to decompose both spaces into finitely many triangles, and they are related by adding/collapsing triangles. It follows that certain spaces can be told apart by string topology. Moreover, we obtain that (part of) string topology can be used to compute Whitehead torsion, which is the invariant associated to simple homotopy theory.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

The main result is a computation of the string topology for lens spaces which shows that the loop coproduct can indeed tell apart two homotopy-equivalent lens spaces. The result is currently submitted to a research journal.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

The project suggests that there is a deep connection between string topology and the Goodwillie-Weiss approximation of manifolds, and along the way showed that there is a connection of both of them to simple homotopy theory. However, the precise connection is still not well understood and subject to further investigation.

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