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Old Drafts
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Brunnian braids.
We
record some observation about Brunnian braids on
the 2-sphere, which lead to the conclusion that
the the cyclic group of order k-1 acts on
the homotopy group \pi_{k-1} (S^2), in k
(potentially) different ways. The actions are
constructed using the description of the
homotopy groups of S^2 in terms of Brunnian
braids on the sphere modulo Brunnian braids on
the disk, due to Berrick-Cohen-Wong-Wu. Posted
2014. (Not intended for publication, since there
is no evidence that these actions are
non-trivial...)
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The homotopy limit
problem in stable representation theory and
the geometry of flat connections.
For
a discrete group G, the relationship between
Carlsson's deformation K-theory spectrum and the
complex connective K-theory of BG can be viewed
as a homotopy limit problem in the sense of
Thomason, providing a natural map of ku-algebras
from the deformation K-theory of G to the
function spectrum F(BG, ku). We show that this
map is closely related to the natural maps Hom
(G, U(n)) → Map(BG, BU(n)), and that it agrees
on homotopy groups with the topological
Atiyah-Segal map introduced by Baird--Ramras
(arXiv:1206.3341). Using results of T. Lawson,
we study this map for products of surface groups
and for crystallographic groups, and give
applications to questions about families of flat
bundles and spaces of flat connections. Posted
9/24/2012
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The spectral
sequence of a tower of cofibrations and
the Lawson spectral sequence.
In this note, we discuss the spectral
sequence associated to a sequence of
cofibrations of spectra, and a resulting
spectral sequence due to Lawson involving
spaces of irreducible representations. We
also give a brief description of an example
of this spectral sequence for surface
groups. (This may be exapanded at some point
in the future - in particular, I hope to
eventually figure out how to draw the
spectral sequence diagrams in LaTex
necessary to explain the example...) Posted
3/27/2012
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An older
version of my paper on
crystallographic groups contained some
information that ended up not being
needed in the final version, but may
still be of some interest.
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Homotopy
invariance of deformation K-theory.
Posted
10/19/2006.
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