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: 2015-09-28
Revised: 2015-12-21
Accepted: 2016-03-24
Published online: 2016-10-04
DOI: https://doi.org/10.5802/aif.3074
Classification: 55P65,  55U35,  55P35,  55P40,  18A30
Keywords: homotopy excision, cellular inequality, total fiber, homotopy localization
     author = {Chach\'olski, Wojciech and Scherer, J\'er\^ome and Werndli, Kay},
     title = {Homotopy excision and cellularity},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     pages = {2641-2665},
     doi = {10.5802/aif.3074},
     language = {en},
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/item/AIF_2016__66_6_2641_0/

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

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

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

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

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

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

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

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

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

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

[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., Tome 16 (2016) no. 1, pp. 325-395 | Article

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

[14] Farjoun, Emmanuel Dror Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, Tome 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.) Tome 265, Amer. Math. Soc., Providence, RI, 2000, pp. 27-39 | Article

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

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

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

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

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

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

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

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