Geodesic
Algebraic models of Poincaré embeddings
Lambrechts, Pascal1 ; Stanley, Donald2
1Institut Mathématique, Université de Louvain, 2, chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
2Department of Mathematics and Statistics, University of Regina, College West 307.14, Regina, Saskatchewan S4S 0A2, Canada
Algebraic and Geometric Topology, Tome 5 (2005) no. 1, pp. 135-182
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

Let f : P↪W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C := W \ T¯ be its complement. Then W is the homotopy push-out of a diagram C ← ∂T → P. This homotopy push-out square is an example of what is called a Poincaré embedding.

We study how to construct algebraic models, in particular in the sense of Sullivan, of that homotopy push-out from a model of the map f. When the codimension is high enough this allows us to completely determine the rational homotopy type of the complement C ≃ W \ f(P). Moreover we construct examples to show that our restriction on the codimension is sharp.

Without restriction on the codimension we also give differentiable modules models of Poincaré embeddings and we deduce a refinement of the classical Lefschetz duality theorem, giving information on the algebra structure of the cohomology of the complement.

DOI : 10.2140/agt.2005.5.135
Keywords: Poincaré embeddings, Lefschetz duality, Sullivan models

Lambrechts, Pascal  1   ; Stanley, Donald  2

1 Institut Mathématique, Université de Louvain, 2, chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
2 Department of Mathematics and Statistics, University of Regina, College West 307.14, Regina, Saskatchewan S4S 0A2, Canada 
@article{10_2140_agt_2005_5_135,
     author = {Lambrechts, Pascal and Stanley, Donald},
     title = {Algebraic models of {Poincar\'e} embeddings},
     journal = {Algebraic and Geometric Topology},
     pages = {135--182},
     year = {2005},
     volume = {5},
     number = {1},
     doi = {10.2140/agt.2005.5.135},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.135/}
}
TY  - JOUR
AU  - Lambrechts, Pascal
AU  - Stanley, Donald
TI  - Algebraic models of Poincaré embeddings
JO  - Algebraic and Geometric Topology
PY  - 2005
SP  - 135
EP  - 182
VL  - 5
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.135/
DO  - 10.2140/agt.2005.5.135
ID  - 10_2140_agt_2005_5_135
ER  - 
%0 Journal Article
%A Lambrechts, Pascal
%A Stanley, Donald
%T Algebraic models of Poincaré embeddings
%J Algebraic and Geometric Topology
%D 2005
%P 135-182
%V 5
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.135/
%R 10.2140/agt.2005.5.135
%F 10_2140_agt_2005_5_135
Lambrechts, Pascal; Stanley, Donald. Algebraic models of Poincaré embeddings. Algebraic and Geometric Topology, Tome 5 (2005) no. 1, pp. 135-182. doi: 10.2140/agt.2005.5.135

[1] C Boilley, Extensions de modules et dualité de Lefschetz, in preparation

[2] A K Bousfield, V K A M Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976)

[3] G E Bredon, Topology and geometry, 139, Springer (1997)

[4] P Deligne, Théorie de Hodge II, Inst. Hautes Études Sci. Publ. Math. (1971) 5

[5] W G Dwyer, J Spaliński, Homotopy theories and model categories, from: "Handbook of algebraic topology", North-Holland (1995) 73

[6] Y Félix, S Halperin, J C Thomas, Gorenstein spaces, Adv. in Math. 71 (1988) 92

[7] Y Félix, S Halperin, J C Thomas, Differential graded algebras in topology, from: "Handbook of algebraic topology", North-Holland (1995) 829

[8] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, 205, Springer (2001)

[9] C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371

[10] M Hovey, Model categories, 63, American Mathematical Society (1999)

[11] S Halperin, Lectures on minimal models, Mém. Soc. Math. France (N.S.) (1983) 261

[12] S Halperin, J M Lemaire, Notions of category in differential algebra, from: "Algebraic topology—rational homotopy (Louvain-la–Neuve, 1986)", Lecture Notes in Math. 1318, Springer (1988) 138

[13] J F P Hudson, Concordance, isotopy, and diffeotopy, Ann. of Math. (2) 91 (1970) 425

[14] T Kahl, P Lambrechts, L Vandembroucq, Bords homotopiques et modèles de Quillen, Homology, Homotopy Appl. 8 (2006) 1

[15] J R Klein, Poincaré duality spaces, from: "Surveys on surgery theory, Vol. 1", Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 135

[16] J R Klein, Poincaré duality embeddings and fibrewise homotopy theory. II, Q. J. Math. 53 (2002) 319

[17] J R Klein, Embedding, compression and fiberwise homotopy theory, Algebr. Geom. Topol. 2 (2002) 311

[18] P Lambrechts, Cochain model for thickenings and its application to rational LS-category, Manuscripta Math. 103 (2000) 143

[19] P Lambrechts, D Stanley, Examples of rational homotopy types of blow-ups, Proc. Amer. Math. Soc. 133 (2005) 3713

[20] P Lambrechts, D Stanley, The rational homotopy type of configuration spaces of two points, Ann. Inst. Fourier (Grenoble) 54 (2004) 1029

[21] P Lambrechts, D Stanley, The rational homotopy type of a blow-up in the stable case, Geom. Topol. 12 (2008) 1921

[22] P Lambrechts, D Stanley, L Vandembroucq, Embeddings up to homotopy of two-cones in Euclidean space, Trans. Amer. Math. Soc. 354 (2002) 3973

[23] N Levitt, On the structure of Poincaré duality spaces, Topology 7 (1968) 369

[24] W B R Lickorish, An introduction to knot theory, 175, Springer (1997)

[25] J W Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1978) 137

[26] C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, , Springer (1982)

[27] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977)

[28] C T C Wall, Finiteness conditions for CW–complexes, Ann. of Math. (2) 81 (1965) 56

[29] C T C Wall, Classification problems in differential topology IV : Thickenings, Topology 5 (1966) 73

Cité par Sources :