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Tuesday Seminar on Topology
Seminar information archive ～10/19｜Next seminar｜Future seminars 10/20～
| Date, time & place | Tuesday 16:00 - 17:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | IKE Yuichi, KONNO Hokuto, SAKASAI Takuya |
2026/11/10
16:00-17:30 Room #hybrid/056 (Graduate School of Math. Sci. Bldg.)
Pre-registration required. See our seminar webpage.
Minkyu Kim (Korea Institute for Advanced Study)
Symmetric powers and double loop spaces of odd spheres (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Pre-registration required. See our seminar webpage.
Minkyu Kim (Korea Institute for Advanced Study)
Symmetric powers and double loop spaces of odd spheres (JAPANESE)
[ Abstract ]
Functors from free groups to abelian groups form a natural setting in algebra and topology, and Ext-groups between such functors govern, for instance, the stable homology of automorphism groups of free groups with twisted coefficients. Over the rationals, these Ext-groups are by now well understood for a wide range of polynomial functors, thanks to work of Christine Vespa. Gregory Arone recently extended Vespa's results to integral coefficients using a new homotopical approach. One case, however, has remained particularly delicate: the Ext-groups from the abelianization functor ab to its symmetric powers S^n. These vanish rationally, but in recent joint work with Vespa, we determined Ext^1 and Ext^2 and found that they contain substantial torsion.
In this talk, I will present an unexpected result obtained recently in joint work with Arone and Vespa: the integral Ext-groups Ext*(ab, S^n) can be expressed in terms of the homology of the double loop space of a sphere of odd dimension at least 3. This identification not only determines the groups in all degrees, but also connects them to the homology of braid groups via classical results in the topology of configuration spaces.
[ Reference URL ]Functors from free groups to abelian groups form a natural setting in algebra and topology, and Ext-groups between such functors govern, for instance, the stable homology of automorphism groups of free groups with twisted coefficients. Over the rationals, these Ext-groups are by now well understood for a wide range of polynomial functors, thanks to work of Christine Vespa. Gregory Arone recently extended Vespa's results to integral coefficients using a new homotopical approach. One case, however, has remained particularly delicate: the Ext-groups from the abelianization functor ab to its symmetric powers S^n. These vanish rationally, but in recent joint work with Vespa, we determined Ext^1 and Ext^2 and found that they contain substantial torsion.
In this talk, I will present an unexpected result obtained recently in joint work with Arone and Vespa: the integral Ext-groups Ext*(ab, S^n) can be expressed in terms of the homology of the double loop space of a sphere of odd dimension at least 3. This identification not only determines the groups in all degrees, but also connects them to the homology of braid groups via classical results in the topology of configuration spaces.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
