KTH/SU
Mathematics
Colloquium
March 22,
2006
Rainer Vogt,
University of
Osnabrueck
Operads,
Interchange, and Iterated Loop Space
Structures
ABSTRACTOperads
encode certain algerbraic structures. Originally
they were introduced to study iterated loop
spaces. The algebraic structure of an n-fold
space \Omega^nX is encoded by a so-called
E_n-operad. Each n-fold loop space has an
E_n-structure and each connected space with an
E_n-structure is of the weak homotopy type of an
n-fold loop space.
Now \Omega^nX has n interchanging single loop
space structures. So a space having an
E_n-structure should have n interchanging
E_1-structures and a space having an E_k- and an
E_l-structure which interchange morally should be
an E_{k+l}-space.
The talk deals with this questioin. We recall the
notions of an operad, of an E_n-structure and of
the interchange of two algebraic structures.
We then address the above mentioned problem. If
time allows we appply the obtained results to
show that the topological Hochschild homology
THH(R) of an E_{n+1}-ring spectrum is an E_n-ring
spectrum and hence has an interesting
multiplicative structure if n>=1.
Remarks. 1) The result about THH(R) was also
obtained by Maria Basterra and Mike Mandell using
different methods.
2) The talk is about joint work with Zig
Fiedorowicz and Roland Schwänzl.