Loday Construction Calculations and Stability

Ayelet Lindenstrauss (Indiana University)

Monday 11th January 16:00-17:00 Online

Abstract

If k is a commutative ring and A is a commutative k-algebra, one can
define a functor from the category of finite sets and functions between
them into the category of commutative rings sending a set into a tensor
product over k of one copy of A for each element of the set.  Functions
of finite sets give functions of these tensor products by just sending
the elements of A over points in the source to lie over the images of
those points, and then multiplying for every point in the target set all
the elements which are now over it.

This method can be used to extend the procedure to simplicial sets, and
get the Loday construction.  It can also be done when k is a commutative
ring spectrum and A is a commutative algebra over it.  It turns out by
general theory that the homotopy groups of the Loday construction on a
simplicial set are homotopy invariants of the realization of the
simplicial set.  And it turns out that the standard definition of
Hochschild homology is exactly the Loday construction of the algebra A
on the minimal model of the circle S^1.  This places Hochschild (and
topological Hochschild) homology calculations in a broader context, and
I will talk about cases where we know how to calculate the homotopy
groups of Loday constructions over general spheres.  Calculations over
higher tori are of interest, as well, for example as targets of iterated
Dennis trace maps from iterated algebraic K-theory, but harder.  I will
discuss stable extensions: pairs of k and A where if there are two
simplicial sets X and Y so that the Loday constructions over their
suspensions have isomorphic homotopy groups, the Loday constructions
over X and Y do, as well.  There is no reason for stability to hold in
general---there are many examples where it does not---but when it does,
it lets us reduce calculations over tori to calculations on spheres.

This is a joint work with Birgit Richter and also Irina Bobkova, Bjorn Dundas, Gemma
Halliwell, Alice Hedenlund, Eva Hoening, Sarah Klanderman, Kate Poirier,
Inna Zakharevich, and Foling Zou in various combinations.



The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/91412568415 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).


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