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Vigleik Angeltveit
Homotopy Associative Multiplications and Homotopy Symmetric Bimodules
Abstract:
In algebraic topology, we usually consider maps between spaces as
"equal" if they are homotopic. So what happens if we consider a
multiplication on a space which is not strictly associative, but only
associative up to homotopy? A typical example is the multiplication on
a loop space X=\Omega Y given by composition of loops.
In this talk, I will introduce some higher coherence conditions such a
multiplication has to satisfy to be considered "good", namely maps K_n
x X^n --> X where K_n is the n'th Stasheff associahedron, a certain
polyhedron homeomorphic to a disk. If such maps exist, we can (under
some technical assumptions) replace X by a homotopy equivalent space
X' with X'=\Omega Y for some Y.
Next I will say what it means for a space M to be a bimodule over X.
In addition to maps X x M --> M and M x X --> M we also need maps
involving the K_n's. Finally, I will give one interpretation of what
it means for M to be symmetric (=left and right action agree) up to
homotopy. This requires another family of polyhedra called cyclohedra.
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