| | |
Summary: THE INTEGRAL COHOMOLOGY OF THE GROUP OF LOOPS
CRAIG JENSEN, JON MCCAMMOND, AND JOHN MEIER
1. introduction
Let Ln be a collection of n unknotted, unlinked circles in 3-space, and let Pn be
the group of motions of Ln where each circle ends up back at its original position.
This group was introduced in the PhD thesis of David Dahm, a student of Ralph
Fox, and was later studied by various authors, notably Deborah Goldsmith. Alan
Brownstein and Ronnie Lee succeeded in computing H2
(Pn, Z) in [6], and at the
end of their paper conjecture a presentation for the algebra H
(Pn, Z). Further
evidence for this conjecture came when the cohomological dimension was computed
(cd(Pn) = n-1) by Collins in [8], and when the Euler characteristic was computed
((Pn) = (1-n)n-1
) in [13] (see also [11]). Here we establish the Brownstein-Lee
Conjecture. As our argument is a mixture of spectral sequences and combinatorial
identities, it seems that Birman was quite prescient in her Mathematical Review of
the Brownstein-Lee paper: "The combinatorics of the cohomology ring appears to
be rich, and the attendant geometric interpretations are very pleasing."
Because 1(S3
|
