On Combinatorial Descriptions of Homotopy Groups of Certain Spaces
This is the revised version of the paper title "On Combinatorial Descriptions
of Homotopy Groups of K(ss; 1)" on the archive. It is significantly changed
from the version which has been on the archive, including adding some new
results.
Jie Wu
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104
USA
jiewu@math.upenn.edu
Included file: newsimplicialgroup_1.dvi
Abstract:
1) We give combinatorial groups which occur naturally for which the homotopy
groups of the suspension of $K(\pi,1)$ for general $\pi$ are the centers.
[Theorem 1.5].
2) We give explicit groups which occur naturally for which all of the
homotopy groups of the 3-sphere are the centers [Theorem 1.2].
3) Furthermore we give an explicit finitely presented nilpotent group for which
the (general) higher homotopy group of the 3-sphere is the torsion of the
center [Proposition 4.9]. In other words, there are explicit finite
2-complexes of which the (general) higher homotopy groups of the $3$-sphere
are the torsion of the centers of the fundamental groups.
4) Our descriptions are NOT yet calculations of higher homotopy groups. But it
is expected to have uses of computer for studying the centers of these
combinatorial groups.
5) We have a group theoretical description of the torsion of homotopy groups of
any simply connected space [Theorem 2.22]. We do not yet have an explicit
combinatorial description of homotopy groups of higher spheres.
6) Our description allows us to compute the homotopy groups of Cohen's
construction of the 1-sphere (the space that is considerable useful in
mathematical physics and deformation theory) and to reprove Milnor's
unpublished example on Moore's problem.