Periodic Reporting for period 1 - StringFrob (String topology and homotopy Frobenius algebras)
Berichtszeitraum: 2021-05-01 bis 2023-04-30
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.
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.
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.