Homotopy excision and cellularity
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2641-2665.

Consider a push-out diagram of spaces CAB, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers–Massey Theorem.

Considérons un diagramme d’espaces CAB, construisons le push-out homotopique, puis le pull-back homotopique du diagramme obtenu en oubliant l’objet initial A. Nous comparons la différence entre A et ce pull-back homomotopique. Cette différence est mesurée en termes des fibres homotopiques des applications originales. En restreignant notre attention sur la connectivité de ces applications nous obtenons la version classique du Théorème de Blakers–Massey.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3074
Classification: 55P65, 55U35, 55P35, 55P40, 18A30
Keywords: homotopy excision, cellular inequality, total fiber, homotopy localization
Mot clés : excision homotopique, inégalité cellulaire, fibre totale, localisation homotopique

Chachólski, Wojciech 1; Scherer, Jérôme 2; Werndli, Kay 2

1 Department of Mathematics KTH Stockholm Lindstedtsvägen 25 10044 Stockholm (Sweden)
2 Department of Mathematics EPFL Lausanne Station 8 1015 Lausanne (Switzerland)
@article{AIF_2016__66_6_2641_0,
     author = {Chach\'olski, Wojciech and Scherer, J\'er\^ome and Werndli, Kay},
     title = {Homotopy excision and cellularity},
     journal = {Annales de l'Institut Fourier},
     pages = {2641--2665},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     doi = {10.5802/aif.3074},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3074/}
}
TY  - JOUR
AU  - Chachólski, Wojciech
AU  - Scherer, Jérôme
AU  - Werndli, Kay
TI  - Homotopy excision and cellularity
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 2641
EP  - 2665
VL  - 66
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3074/
DO  - 10.5802/aif.3074
LA  - en
ID  - AIF_2016__66_6_2641_0
ER  - 
%0 Journal Article
%A Chachólski, Wojciech
%A Scherer, Jérôme
%A Werndli, Kay
%T Homotopy excision and cellularity
%J Annales de l'Institut Fourier
%D 2016
%P 2641-2665
%V 66
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3074/
%R 10.5802/aif.3074
%G en
%F AIF_2016__66_6_2641_0
Chachólski, Wojciech; Scherer, Jérôme; Werndli, Kay. Homotopy excision and cellularity. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2641-2665. doi : 10.5802/aif.3074. https://aif.centre-mersenne.org/articles/10.5802/aif.3074/

[1] Biedermann, Georg; Chorny, Boris; Röndigs, Oliver Calculus of functors and model categories, Adv. Math., Volume 214 (2007) no. 1, pp. 92-115 | DOI

[2] Blakers, A. L.; Massey, W. S. The homotopy groups of a triad. II, Ann. of Math. (2), Volume 55 (1952), pp. 192-201 | DOI

[3] Bousfield, A. K. Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc., Volume 7 (1994) no. 4, pp. 831-873 | DOI

[4] Brown, Ronald; Loday, Jean-Louis Homotopical excision, and Hurewicz theorems for n-cubes of spaces, Proc. London Math. Soc. (3), Volume 54 (1987) no. 1, pp. 176-192 | DOI

[5] Chachólski, Wojciech Closed classes, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994) (Progr. Math.), Volume 136, Birkhäuser, Basel, 1996, pp. 95-118 | DOI

[6] Chachólski, Wojciech On the functors CW A and P A , Duke Math. J., Volume 84 (1996) no. 3, pp. 599-631 | DOI

[7] Chachólski, Wojciech Desuspending and delooping cellular inequalities, Invent. Math., Volume 129 (1997) no. 1, pp. 37-62 | DOI

[8] Chachólski, Wojciech A generalization of the triad theorem of Blakers-Massey, Topology, Volume 36 (1997) no. 6, pp. 1381-1400 | DOI

[9] Chachólski, Wojciech; Farjoun, Emmanuel Dror; Flores, Ramón; Scherer, Jérôme Cellular properties of nilpotent spaces, Geom. Topol., Volume 19 (2015) no. 5, pp. 2741-2766 | DOI

[10] Chachólski, Wojciech; Scherer, Jérôme Homotopy theory of diagrams, Mem. Amer. Math. Soc., Volume 155 (2002) no. 736, x+90 pages | DOI

[11] Ching, Michael; Harper, John E. Higher homotopy excision and Blakers-Massey theorems for structured ring spectra (2014) (preprint, http://arxiv.org/abs/1402.4775)

[12] Dotto, Emanuele; Moi, Kristian Homotopy theory of G–diagrams and equivariant excision, Algebr. Geom. Topol., Volume 16 (2016) no. 1, pp. 325-395 | DOI

[13] Ellis, Graham; Steiner, Richard Higher-dimensional crossed modules and the homotopy groups of (n+1)-ads, J. Pure Appl. Algebra, Volume 46 (1987) no. 2-3, pp. 117-136 | DOI

[14] Farjoun, Emmanuel Dror Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, 1622, Springer-Verlag, Berlin, 1996, xiv+199 pages

[15] Farjoun, Emmanuel Dror Two completion towers for generalized homology, Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999) (Contemp. Math.), Volume 265, Amer. Math. Soc., Providence, RI, 2000, pp. 27-39 | DOI

[16] Goodwillie, Thomas G. Calculus. II. Analytic functors, K-Theory, Volume 5 (1991/92) no. 4, pp. 295-332 | DOI

[17] Groth, Moritz Derivators, pointed derivators and stable derivators, Algebr. Geom. Topol., Volume 13 (2013) no. 1, pp. 313-374 | DOI

[18] Klein, John R.; Peter, John W. Fake wedges, Trans. Amer. Math. Soc., Volume 366 (2014) no. 7, pp. 3771-3786 | DOI

[19] Mather, Michael Pull-backs in homotopy theory, Canad. J. Math., Volume 28 (1976) no. 2, pp. 225-263 | DOI

[20] Munson, Brian A.; Volić, Ismar Cubical Homotopy Theory, new mathematical monographs, 28, Cambridge University Press, 2015, xv+631 pages

[21] Puppe, Volker A remark on “homotopy fibrations”, Manuscripta Math., Volume 12 (1974), pp. 113-120 | DOI

[22] Weiss, Michael Orthogonal calculus, Trans. Amer. Math. Soc., Volume 347 (1995) no. 10, pp. 3743-3796 | DOI

[23] Whitehead, George W. Elements of homotopy theory, Graduate Texts in Mathematics, 61, Springer-Verlag, New York-Berlin, 1978, xxi+744 pages

Cited by Sources: