220241
20190509132558.0
doi
10.5075/epfl-thesis-7127
urn
urn:nbn:ch:bel-epfl-thesis7127-4
nebis
10687447
THESIS
eng
7127
Cellular Homotopy Excision
2016
Lausanne
EPFL
2016
288
Theses
Prof. Marc Troyanov (président) ; Prof. Kathryn Hess Bellwald, Dr Jérôme Scherer (directeurs) ; Prof. Tamás Hausel, Prof. Wojciech Chachólski, Prof. Emmanuel Dror Farjoun (rapporteurs)
There is a classical "duality" between homotopy and homology groups in that homotopy groups are compatible with homotopy pullbacks (every homotopy pullback gives rise to a long exact sequence in homotopy), while homology groups are compatible with homotopy pushouts (every homotopy pushout gives rise to a long exact sequence in homology). This last statement is sometimes referred to as the Mayer-Vietoris or excision axiom. The classical Blakers-Massey theorem (or homotopy excision theorem) asks to what extent the excision property for homotopy pushouts remains true if we replace homology groups by homotopy groups and gives a range in which the excision property holds. It does so by estimating the connectivity of a certain comparison map, which is a rather crude measure, as it is just a single number. Since connectivity is a special case of a cellular inequality, the hope is that there is a stronger statement hidden behind the connectivity result in terms of such inequalities. This process of generalising the homotopy excision theorem has been initiated by Chachólski in the 90s, where he proved a more general version for homotopy pushout squares. The caveat was that one had to suspend the comparison map in question first and the goal of our project -- which we obtained -- was to lose this suspension and then move on to cubical diagrams, rather than squares. To do so, there are a few basic ingredients that are necessary. We first talk about our abstract approach to derived functors, then construct left Bousfield localisations of combinatorial model categories and finally, generalise the foundational concepts in the theory of closed classes to non-connected spaces.
Homotopy excision
Bousfield localisation
Bousfield classes
Cellular classes
Closed classes
Diagrams of spaces
Derived functors
245810
Werndli, Kay Remo
216064
240499
Hess Bellwald, Kathryn
dir.
105396
243126
Scherer, Jérôme
dir.
144617
6449136
http://infoscience.epfl.ch/record/220241/files/EPFL_TH7127.pdf
n/a
n/a
252139
UPHESS
U10968
oai:infoscience.tind.io:220241
thesis
thesis-bn2018
DOI
SV
DOI2
GLOBAL_SET
108898
108898
108898
108898
SB
MATHGEOM
EDMA
GR-HE
2016-8-12
2016
7127/THESES
EPFL
PUBLISHED
THESIS