Abstract
The purpose of this paper is to calculate the rational cohomology \({H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}\) of the free loop space for a simply connected closed 4-manifold X. We use minimal models, so the starting point is the cohomology algebra \({H^{\ast}(X; \mathbb{Q})}\) which depends only on the second Betti number b 2 and the signature of X itself. Calculations of \({H^{\ast}(X^{{S}^{1}} ; \mathbb{Q})}\) for b 2 ≤ 2 are known. We study the case b 2 > 2. We obtain an explicit formula for Poincaré series of the space \({X^{{S}^{1}}}\), with the second Betti number b 2 as a parameter.
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References
Babenko I (1980). Analytic properties of Poincaré series of a loop space. Mat. Zametki 27: 751–765
Babenko. I. (1979). On real homotopy properties of complete intersections. Izv. Akad. Nauk SSSR Ser. Mat. 43: 1004–1024
D. W. Barnes, Spectral sequence constructors in algebra and topology. Mem. Amer. Math. Soc. 53 (1985), viii+174.
R. Bott and L. Tu, Differential Forms in Algebraic Topology. Grad. Texts in Math. 82, Springer-Verlag, New York, 1982.
M. Chas and D. Sullivan, String topology. Preprint, arXiv:math/9911159.
R. Cohen, J. Jones and J. Yan, The loop homology algebra of spheres and projective spaces. In: Categorical Decomposition Techniques in Algebraic Topology (Isle of Skye, 2001), Progr. Math. 215, Birkhäuser, Basel, 2004, 77–92.
Dress A. (1967). Zur Spectralsequenz von Faserungen. Invent. Math. 3: 172–178
Y. Félix, S. Halperin and J. C. Thomas, Rational Homotopy Theory. Grad. Texts in Math. 205, Springer-Verlag, New York, 2001.
Y. Félix, J. C. Thomas and M. Vigué-Poirrier, Loop homology algebra of a closed manifold. Preprint, arXiv:math/0203137.
A. T. Fomenko, D. B. Fuchs and V. L. Gutenmacher, Homotopic Topology. Akademiai Kiado, Budapest, 1986.
Ph. Griffiths and J. Morgan, Rational homotopy theory and differential forms. Progress in Mathematics 16, Birkh¨auser Boston, Mass., 1981.
K. Kuribayashi and T. Yamaguchi, The cohomology algebra of certain free loop spaces. Fund. Math. 154 (1997), 57–73.
J. McCleary, A User’s Guide to Spectral Sequences. Cambridge Stud. Adv. Math. 58, Cambridge Univesity Press, Cambridge, 2001.
J. Milnor and J. C. Moore, On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965), 211–264.
Neisendorfer J. (1978). Formal and coformal spaces. Illinois J. Math. 22: 565–580
J. Neisendorfer, The rational homotopy groups of complete intersections. Illinois J. Math. 23 (1979), 175–182.
A. Yu. Onishchenko, On the center of the rational cohomology algebra of some loop spaces with respect to the Pontryagin product. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 208 (2008), 28–33.
A. Onishchenko and Th. Popelensky, On the equivalence of some spectral sequences for Serre fibrations. Sb. Math. 202 (2011), 547–570.
J. P. Serre, Homologie singulière des espaces fibrés. Applications. Ann. of Math. (2) 54 (1951), 425–505.
M. Vigué-Poirrier and D. Burghelea, A model for cyclic homology and algebraic K-theory of 1-connected topological spaces. J. Differential Geom. 22 (1985), 243–253.
Vigué-Poirrier M. and Sullivan D (1976). The homology theory of the closed geodesic problem. J. Differential Geom. 11: 633–644
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Onishchenko, A.Y., Popelensky, T.Y. Rational cohomology of the free loop space of a simply connected 4-manifold. J. Fixed Point Theory Appl. 12, 69–92 (2012). https://doi.org/10.1007/s11784-013-0100-0
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DOI: https://doi.org/10.1007/s11784-013-0100-0