×

The rational homotopy type of a blow-up in the stable case. (English) Zbl 1153.55010

Let \(f\colon V\to W\) a smooth embedding of closed manifolds of codimension \(2k\) (\(k>1\)) such that the normal bundle \(\nu\) of this embedding has the structure of a \(k\)-complex bundle \(\nu\colon \mathbb C^k\to E\buildrel{\pi}\over{\to}V\).
In this paper, the authors construct a Sullivan model of the projectivization of \(\nu\), \(\mathbb CP(k-1)\hookrightarrow P\nu\buildrel{\pi'}\over\to V\), out of a model of the embedding and the Chern classes of \(\nu\), \(c_i(\nu)\in H^{2i}(V;\mathbb Z)\).
In fact, if \(A\) is a particular common model of \(V\) and of a compact tubular neighborhood of \(V\) inside \(W\), a model of \(P\nu \) is given by \((A\otimes\Lambda (x,z),D)\), in which \(x\) has degree \(2\), \(y\) has degree \(2k-1\), \(Dx=0\) and \(Dz=\sum_{i=0}^k \gamma_i x^{k-i}\). Here, the \(\gamma_i\)’s denote representatives of the image of the Chern classes through the map \(H^{2i}(V;\mathbb Z)\to H^{2i}(V;\mathbb Q)\).
Next the authors give an interesting description of the blow-up construction of McDuff for symplectic manifolds in this more general setting.
Finally, the results above, interesting in their own rite, are combined to prove the main result: if \(dim\,W\geq 2\,dim\,V+3\) and \(H^1(f)\) is injective, the authors build a Sullivan model of the blow-up \(\widetilde W\) of \(W\) along \(V\). Explicitly, if \(\phi\colon R\to A\) is a model of the embedding \(f\colon V\to W\), the model of \(\widetilde W\) has the form \(B(R,A)=(R\oplus A\otimes \Lambda^+(x,z),D)\) in which the algebra structure is given by the one in \(R\) and \(A\otimes \Lambda^+(x,z)\), and by the \(R\)-module structure on \(A\otimes \Lambda^+(x,z)\) via the morphism \( \phi\). The differential also involves the previous constructions.
This deep result proves in particular that, if \(W\) is simply connected, the rational homotopy type of the blow-up only depends on the rational homotopy class of the embedding and on the Chern classes of the normal bundle \(\nu\).
Interesting applications of these results to the behavior of blow-ups of complex projective spaces are also included.

MSC:

55P62 Rational homotopy theory
14F35 Homotopy theory and fundamental groups in algebraic geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D05 Symplectic manifolds (general theory)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] I K Babenko, I A Taĭmanov, On the existence of nonformal simply connected symplectic manifolds, Uspekhi Mat. Nauk 53 (1998) 225 · Zbl 0972.53049
[2] R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Math. 82, Springer (1982) · Zbl 0496.55001
[3] A K Bousfield, V K A M Gugenheim, On \(\mathrm{PL}\) de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976) · Zbl 0338.55008
[4] G E Bredon, Topology and geometry, Graduate Texts in Math. 139, Springer (1997) · Zbl 0934.55001
[5] P Deligne, P Griffiths, J Morgan, D Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975) 245 · Zbl 0312.55011
[6] W G Dwyer, J Spaliński, Homotopy theories and model categories, North-Holland (1995) 73 · Zbl 0869.55018
[7] Y Félix, S Halperin, J C Thomas, Gorenstein spaces, Adv. in Math. 71 (1988) 92 · Zbl 0659.57011
[8] Y Félix, S Halperin, J C Thomas, Differential graded algebras in topology (editor I James), North-Holland (1995) 829 · Zbl 0868.55016
[9] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer (2001)
[10] S Gitler, The cohomology of blow ups, Bol. Soc. Mat. Mexicana \((2)\) 37 (1992) 167 · Zbl 0836.57018
[11] P Griffiths, J Harris, Principles of algebraic geometry, Pure and Applied Math., Wiley-Interscience (1978) · Zbl 0408.14001
[12] M Gromov, Partial differential relations, Ergebnisse der Math. und ihrer Grenzgebiete (3) 9, Springer (1986) · Zbl 0651.53001
[13] S Halperin, Lectures on minimal models, Mém. Soc. Math. France \((\)N.S.\()\) (1983) 261 · Zbl 0536.55003
[14] P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies 15, Notas de Matemática (55), North-Holland Publishing Co. (1975) · Zbl 0323.55016
[15] P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monographs 99, Amer. Math. Soc. (2003) · Zbl 1017.55001
[16] M Hovey, Model categories, Math. Surveys and Monographs 63, Amer. Math. Soc. (1999) · Zbl 0909.55001
[17] P Lambrechts, D Stanley, Blowing up in \(\mathbb{CP}(n)\) preserves non-formality, in preparation
[18] P Lambrechts, D Stanley, Algebraic models of Poincaré embeddings, Algebr. Geom. Topol. 5 (2005) 135 · Zbl 1114.55007
[19] P Lambrechts, D Stanley, Examples of rational homotopy types of blow-ups, Proc. Amer. Math. Soc. 133 (2005) 3713 · Zbl 1077.55007
[20] D McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984) 267 · Zbl 0567.53031
[21] D McDuff, D Salamon, Introduction to symplectic topology, Oxford Math. Monographs, The Clarendon Oxford University Press (1998) · Zbl 0844.58029
[22] J W Milnor, J D Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton University Press (1974) · Zbl 0298.57008
[23] D G Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer (1967) · Zbl 0168.20903
[24] V Rao, Nilpotency of homotopy pushouts, Proc. Amer. Math. Soc. 87 (1983) 335 · Zbl 0513.55009
[25] Y Rudyak, A Tralle, On Thom spaces, Massey products, and nonformal symplectic manifolds, Internat. Math. Res. Notices (2000) 495 · Zbl 0972.53052
[26] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977) · Zbl 0374.57002
[27] R M Switzer, Algebraic topology-homotopy and homology, Die Grundlehren der math. Wissenschaften 212, Springer (1975) · Zbl 0305.55001
[28] W P Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976) 467 · Zbl 0324.53031
[29] D Tischler, Closed \(2\)-forms and an embedding theorem for symplectic manifolds, J. Differential Geometry 12 (1977) 229 · Zbl 0386.58001
[30] A Tralle, J Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Math. 1661, Springer (1997) · Zbl 0891.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.