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Rational Homotopy Theory in Mathematics and Physics

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Rational Homotopy Theory in Mathematics and Physics Mathematisches Forschungsinstitut Oberwolfach Report No. 18/2011 DOI: 10.4171/OWR/2011/18 Arbeitsgemeinschaft: Rational Homotopy Theory in Mathematics and Physics Organised by John Oprea, Cleveland Daniel Tanr´e, Lille April 2nd – April 8th, 2011 Abstract. This Arbeitsgemeinschaft focused on the interplay among ratio- nal homotopy theory, differential geometry and the physics of string topology. The talks centered on one hand on how geometry and string topology make use of rational homotopy methods, elicit new questions in rational homotopy and lead to the development of new rational homotopy structures reflecting their Natures; and on the other hand on how rational homotopy theory has given concrete results in geometry. Mathematics Subject Classification (2000): 18G10, 18G55, 53C22, 53C55, 53D35, 55C05, 55G30, 55P62, 57A65, 57B10, 57T20. Introduction by the Organisers In [9], Sullivan defined tools and models for rational homotopy inspired by already existing geometrical objects. Moreover, he gave an explicit dictionary between his minimal models and spaces, and this facility of transition between algebra and topology has created many new topological and geometrical theorems in the last 30 years. ∗ ∗ R When de Rham proved that H (ADR(M)) ∼= H (M; ) for the differential algebra of differential forms ADR(M) on a manifold M, it immediately provided a link between the analysis on and the topology of the manifold. Sullivan suggested that even within the world of topology, there is more topological information in the de Rham algebra of M than simply the real cohomology. In the de Rham algebra, there is information contained in two different entities: the product of forms, which tells us how two forms can be combined together to 994 Oberwolfach Report 18/2011 give a third one and the exterior derivative of a form. In a model, we kill the infor- mation coming from the product structure by considering free algebras V (in the commutative graded sense) where V is an R-vector space. This pushes∧ the corre- sponding information into the differential and into V where it is easier to detect. More precisely, we look for a cdga (for commutative differential graded algebra) free as a commutative graded algebra ( V, d) and a morphism ϕ: ( V, d) ADR(M) inducing an isomorphism in cohomology.∧ ∧ → For instance, if G is a compact connected Lie group, there exists a sub-differential algebra of bi-invariant forms, ΩI (G), inside the de Rham algebra ADR(G), such -that the canonical inclusion ΩI (G) ֒ ADR(G) induces an isomorphism in coho mology. This is the prototype of the→ process for models: namely, we look for a simplification of the de Rham algebra with an explicit differential morphism MM M ADR(M) inducing an isomorphism in cohomology, exactly as bi-invariant formsM → do in the case of a compact connected Lie group. The first question is, can one build such a model for any manifold? The answer is yes for connected manifolds and in fact, there are many ways to do this. So, we describe a standard way, which is called minimal, and which is defined by requiring that the differentials of elements of V have no linear terms. Once we have this minimal model (which is unique up to isomorphism), we may ask what geometrical invariants can be detected in it. In fact, there is a functor from algebra to geometry that, together with forms, creates a dictionary between the algebraic and the geometrical worlds. But for this we have to work over the rationals and not over the reals. As a consequence, we have to replace the de Rham algebra by other types of forms. This new construction is very similar to the de Rham algebra and allows the extension of the usual theory from manifolds to simplicial sets (or topological spaces), which is a great advantage. Denote by APL(X) this analogue of the de Rham algebra for a simplicial set X. Since the minimal model construction also works perfectly well over Q, we have the notion of a minimal model X APL(X) of a path connected space X. Conversely,M → from a cdga (A, d) we have a topological realization (A, d) and if h i we apply this realization to a minimal model X of a space X (which is nilpotent with finite Betti numbers), then we get a continuousM map X which → hMX i induces an isomorphism in rational cohomology. The space X is what, in homotopy theory, is called a rationalization of X. What musthM bei emphasized in this process is the ability to create topological realizations of any algebraic constructions. Hence, Sullivan’s theory can be seen as a rational version of classical differential geometry. Such theories beg for applications and examples and it is possible to describe models for spheres, homogeneous spaces, biquotients, nilmanifolds, symplectic blow-ups and the free loop space. These models have geometrical applications, for instance, to complex and symplectic manifolds, the closed geodesic problem, cur- vature questions, actions of tori and, more recently, the Chas-Sullivan loop prod- uct. The focus of this Arbeitsgemeinschaft was the relationship between Rational Arbeitsgemeinschaft: Rational Homotopy Theory 995 Homotopy Theory and Geometry with a natural extension to Physics via string topology. Several monographs are devoted to these theories: [1, 4, 5, 6, 10, 11]. The Arbeitsgemeinschaft consisted of 18 talks whose overriding goal was to tie together problems in geometry with rational homotopy theory. For instance, the question of the existence of infinitely many geodesics on a closed manifold was shown to be intimately tied up with the rational homotopy of the free loop space. Similarly, the rational homotopy qualities of a space were shown to depend on special properties of the geodesic flow on the manifold (considered as a Hamiltonian system). Rational homotopy was also shown to be important to understanding the difference between K¨ahler and symplectic manifolds as well as a key ingredient in treating certain “rational” problems about sectional curvature. (References for these items will be given in the following abstracts.) The Arbeitsgemeinschaft was attended by about 50 people, including many young mathematicians who gave excellent talks. No one was lost on the traditional hike. References [1] A. K. Bousfield, and V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc., Memoirs of the American Mathematical Society, vol. 8 no. 179 (1976). [2] M. Cai, A splitting theorem for manifolds of almost nonnegative Ricci curvature, Annals of Global Analysis and Geometry 11 (1993) 373-385. [3] O. Cornea, G. Lupton, J. Oprea and D. Tanr´e, Lusternik-Schnirelmann category, Math- ematical Surveys and Monographs, 103. American Mathematical Society, Providence, RI, 2003. [4] Y. F´elix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer-Verlag, New York (2001). [5] Y. F´elix, J. Oprea, and D. Tanr´e, Algebraic models in geometry, Oxford Graduate Texts in Mathematics, 17, Oxford University Press, Oxford,(2008). [6] P. Griffiths and J. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, 16, Birkh¨auser Boston Mass., (1981). [7] V. Kapovitch, A. Petrunin and W. Tuschmann, Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. of Math. (2) 171 (2010), no. 1, 343373. [8] J. Oprea, Category bounds for nonnegative Ricci curvature manifolds with infinite funda- mental group, Proc. Amer. Math. Soc. 130 (2002), no. 3, 833839. [9] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes´ Sci. Publ. Math., Institut des Hautes Etudes´ Scientifiques. Publications Math´ematiques, 47 (1977) 269–331. [10] D. Tanr´e, Homotopie rationnelle: mod`eles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics 1025, Springer-Verlag, Berlin, (1983). [11] A. Tralle and J. Oprea, Symplectic manifolds with no K¨ahler structure, Lecture Notes in Mathematics, 1661, Springer-Verlag, Berlin, (1997). [12] T. Yamaguchi, Manifolds of almost nonnegative curvature (Russian summary) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 234 (1996), Differ. Geom. Gruppy Li i Mekh. 15-1, 201–260, 265; translation in J. Math. Sci. (New York) 94 (1999), no. 2, 12701310. [13] K. Yano, and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32. Princeton University Press, Princeton, N. J., 1953. Arbeitsgemeinschaft: Rational Homotopy Theory 997 Arbeitsgemeinschaft: Rational Homotopy Theory in Mathemat- ics and Physics Table of Contents Wilderich Tuschmann Fundamentals of Geometry ....................................... 999 Maura Macr`ı Sullivan Models ................................................. 999 Marc Stephan Group Actions ..................................................1001 Stephan Wiesendorf Geodesics and the Free Loop Space I ...............................1003 Varvara Karpova Geodesics and the Free Loop Space II ..............................1005 Nadine Große Geodesic Flows ..................................................1007 Pascal Lambrechts Formality and K¨ahler Manifolds ...................................1009 Joana Cirici Spectral Sequences and Models ....................................1012 Victor Turchin Formality and Symplectic Manifolds I ..............................1015 Achim Krause Formality and Symplectic Manifolds II .............................1018 Benjamin Matschke Curvature and Rational Homotopy I – Many manifolds with bounded curvature and diameter ..........................................1019 Sebastian Goette Curvature II ..................................................
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