Maps of degree one, LS category and higher topological complexities
DOI:
https://doi.org/10.12775/TMNA.2021.051Keywords
Lusternik-Schnirelmann category, sectional category, topological complexityAbstract
In this paper, we study the relation between the Lusternik-Schnirelmann category and the topological complexity of two closed oriented manifolds connected by a degree one map.References
J.F. Adams, Lectures on Exceptional Lie groups, (J.P. May, fireword; Z. Mahmud and M. Mimura, eds.) Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996, xiv+122 pp.
M. Arkowitz, Introduction to Homotopy Theory, Universitext, Springer, New York, 2011.
I. Basabe, J. González, Y.B. Rudyak and D. Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topology 14 (2014), 2103–2124.
M. Bayeh and S. Sarkar, Some aspects of equivariant LS-category, Topology Appl. 196 (2015), 133–154.
A. Borel, Sur l’homologie et la cohomologie des groupes de Lie compact connexes, Amer. J. Math. 76 (1954), 273–342.
W. Browder, Surgery on Simply-Connected Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer–Verlag, New York, Heidelberg, 1972.
O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, vol. 103, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2003.
M. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2.
A.N. Dranishnikov, The LS category of the product of lens spaces, Algebr. Geom. Topol. 15 (2015), no. 5, 2985–3010.
A.N. Dranishnikov, M. Katz and Y. Rudyak, Small values of the Lusternik–Schnirelman category for manifolds, Geom. Topol. 12 (2008), no. 3, 1711–1727.
A.N. Dranishnikov and Y. Rudyak, On the Berstein–Svarc theorem in dimension 2, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 407–413.
A.N. Dranishnikov and R. Sadykov, On the LS-category and topological complexity of a connected sum, Proc. Amer. Math. Soc. 147 (2019), no. 5, 2235–2244.
A.N. Dranishnikov and J. Scott, Surgery approach to Rudyak’s conjecture, Topology Appl. 311 (2022), paper no. 107956, 7 pp.
E. Dyer, Cohomology Theories, Mathematics Lecture Note Series, W.A. Benjamin, Inc., New York, Amsterdam, 1969.
E. Fadell and S. Husseini, Category weight and Steenrod operations, Papers in honor of José Adem, Bol. Soc. Mat. Mexicana (2) 37 (1992), no. 1–2, 151–161.
M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29 (2003), no. 2, 211–221.
M. Farber, Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zurich, 2008.
M. Farber and M. Grant, Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3339–3349.
M. Grant, G. Lupton and J. Oprea, Spaces of topological complexity one, Homology Homotopy Appl. 15 (2013), no. 2, 73–81.
J. Hempel, 3-Manifolds, Ann. of Math. Studies, vol. 86, Princeton Univ. Press, Princeton, New Jersey, 1976.
N. Iwase and M. Mimura, L–S categories of simply-connected compact simple Lie groups of low rank, Categorical Decomposition Techniques in Algebraic Topology (Isle of Skye, 2001), Progr. Math., vol. 215, Birkhäuser, Basel, 2004, 199–212.
I. James, On category, in the sense of Lusternik–Schnirelmann, Topology 17 (1978), no. 4, 331–348.
I. James and J. Whitehead, The homotopy theory of sphere bundles over spheres I, Proc. London Math. Soc. (3) 4 (1954), 196–218.
M. Krasnosel’skı̆, On special coverings of a finite-dimensional sphere, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 961–964 (in Russian).
M. Masuda and T. Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006), no. 3, 711–746.
Y.B. Rudyak, Some remarks on category weight, Preprint, University of Heidelberg, 1996.
Y.B. Rudyak, On Thom spectra, orientability, and cobordism (H. Miller, foreword), Springer Monographs in Mathematics, Springer–Verlag, Berlin, 1998.
Y.B. Rudyak, On category weight and its applications, Topology 38 (1999), no. 1, 37–55.
Y.B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916–920; errata: Topology Appl. 157 (2010), 1118.
Y.B. Rudyak, Maps of degree 1 and Lusternik–Schnirelmann category, Topology Appl. 221 (2017), 225–230.
Y.B. Rudyak and J. Oprea, On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Z. 230 (1999), no. 4, 673–678.
Y.B. Rudyak and S. Sarkar, Maps of degree one, relative LS category and higher topological complexities, arXiv: 2011.13531.
A.S. Schwarz (Švarc), The genus of a fiber space, Amer. Math. Soc. y Transl. 55 (1966), 49–140.
J. Strom, Essential category weight, Preprint, University of Wisconsin–Madison, 1997.
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