Abstract
We show that for closed orientable manifolds the k-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree k that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles of dimension at least two. Additionally, we prove that the stable systolic constant depends only on the image of the fundamental class in a suitable Eilenberg–Mac Lane space. Consequently, the stable k-systolic constant is completely determined by the multilinear intersection form on k-dimensional cohomology.
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References
Babenko, I.K.: Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat. 56(4), 707–751 (1992) (Russian) [Engl. Transl. Russian Acad. Sci. Izv. Math. 41(1), 1–38 (1993)]
Babenko I.K.: Forte souplesse intersystolique de variétés fermées et de polyèdres. Ann. Inst. Fourier (Grenoble) 52(4), 1259–1284 (2002)
Babenko I.K.: Topologie des systoles unidimensionnelles. Enseign. Math. (2) 52(1-2), 109–142 (2006)
Babenko I.K., Katz M.G.: Systolic freedom of orientable manifolds. Ann. Sci. École Norm. Sup. (4(31(6), 787–809 (1998) See arXiv:dg-ga/9707002
Babenko I.K., Katz M.G., Suciu A.I.: Volumes, middle-dimensional systoles, and Whitehead products. Math. Res. Lett. 5(4), 461–471 (1998) See arXiv:dg-ga/9707016
Bangert, V., Katz, M.G.: Stable systolic inequalities and cohomology products. Comm. Pure Appl. Math. 56(7), 979–997. Dedicated to the memory of Jürgen K. Moser, see arXiv:math/0204181 [math.DG] (2003)
Bangert, V., Katz, M.G.: Steven Shnider, and Shmuel Weinberger, E 7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Math. J. (2006, to appear). arXiv:math/0608006 [math.DG]
Berger M.: Du côté de chez Pu. Ann. Sci. École Norm. Sup. 5(4), 1–44 (1972)
Berger M.: A Panoramic View of Riemannian geometry. Springer (2003)
Brunnbauer, M.: Homological invariance for asymptotic invariants and systolic inequalities. Geom. Funct. Anal. (2007, to appear). arXiv:math/0702789 [math.GT]
Croke, C.B., Katz M.G.: Universal volume bounds in Riemannian manifolds, Surveys in differential geometry 2002, vol. VIII, pp. 109–137. Int. Press, Boston, (2003). See arXiv:math/0302248 [math.DG]
Federer H.: Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24(4), 351–407 (1974)
Gromov M.L.: Filling Riemannian manifolds. J. Diff. Geom. 18(1), 1–147 (1983)
Gromov M.L.: Metric Structures for Riemannian and Non-Riemannian Spaces, progress in mathematics, vol. 152. Birkhäuser (1999)
Hamilton M.J.D.: On the conformal systoles of four-manifolds. Manuscr. Math. 121(4), 417–424 (2006) See arXiv:math/0512127 [math.DG]
Hatcher, A.: Algebraic topology, Cambridge University Press, freely available at http://www.math.cornell.edu/~hatcher/AT/ATpage.html (2002)
Katz, M.G.: Systolic Geometry and Topology, Mathematical Surveys and Monographs, vol. 137 (with an appendix by Solomon J.P.). American Mathematical Society (2007)
Katz, M.G., Suciu, A.I.: Volume of Riemannian Manifolds, Geometric Inequalities, and Homotopy Theory, Tel Aviv Topology Conference: Rothenberg Festschrift (1998) Contemp. Math., vol. 231, pp. 113–136. Am. Math. Soc. (1999). See arXiv:math/9810172 [math.DG]
Katz, M.G., Suciu, A.I.: Systolic freedom of loop space. Geom. Funct. Anal. 11(1), 60–73 (2001). (An updated version of the published paper may be found at arXiv:math/0106153 [math.DG])
Sabourau S.: Systolic volume and minimal entropy of aspherical manifolds. J. Diff. Geom. 74(1), 155–176 (2006) See arXiv:math/0603695 [math.DG]
Serre J.-P.: Homologie singulière des espaces fibrés. III. Applications homotopiques. C. R. Acad. Sci. Paris 232, 142–144 (1951)
Sullivan D.: Genetics of homotopy theory and the Adams conjecture. Ann. Math. 100(2), 1–79 (1974)
Whitehead G.W.: Elements of Homotopy Theory. Graduate Texts in Mathematics, vol. 61. Springer, Heidelberg (1978)