STABILISATION DES COMPLEXES CROISES
Orin R. Sauvageot
orin.sauvageot@epfl.ch
Ecole Polytechnique Federale de Lausanne
Institute of Mathematics
This is my PhD thesis in FRENCH, 158 pages. The graphic files
C-tensor-I.eps, pi-delta-4.eps and pi-xc.eps are included in the zip
archives thesis-print.dvi.zip and thesis-screen.dvi.zip.
Abstract
In this doctoral thesis we present a stabilization of the category of
crossed complexes. Our work is motivated by the difficulty one has in
performing algebraic calculations in Boardman's stable homotopy
category, since products and actions are defined only up to homotopy in
the underlying category of spectra, as defined by Bousfield and
Friedlander. To correct this lack of precision, a number of new models
of the stable homotopy category have been developed in which algebraic
constructions are exactly defined. One such model is the category of
symmetric spectra on simplicial sets, the manipulation of which is still
not easy, however.
The idea behind this thesis is to stabilize the category of crossed
complexes, as it is an interesting approximation to the category of
simplicial sets, reflecting certain, though not all, nonabelian
homotopical information concerning simplicial sets. We have stabilized
it according to the procedure codified in Hovey's "Spectra and symmetric
spectra in general model categories".
Stabilization requires that the category of crossed complexes satisfies
certain properties. We have succeeded in proving these properties, in
each case establishing a previously unknown result. For example, we
have shown that it is cofibrantly generated and that it is a symmetric
monoidal model category. Furthermore we have verified that it is a
proper, cellular category. In proving the properness we have answered
an open question posed by Brown and Golasinski. In the course of
establishing these properties we have established a nonabelian version
of the 5-Lemma.
A crossed complex is a generalization of a chain complex of abelian
groups. We have shown, however, that the stabilization of crossed
complexes is homotopy equivalent to that of the category of chain
complexes. On the other hand, the situation of unpointed crossed
complexes is different, and it is very likely that their stabilization
is not that of chain complexes. In order to argue so, we have
constructed an innovative simplicial model of the Hopf map. It remains
then to give a topological meaning to an unpointed stabilization. An
attempt of answer is sketched.