Research Article
Year 2025,
Volume: 54 Issue: 2, 352 - 367, 28.04.2025
Abstract
In this paper, using Sullivan's approach to rational homotopy theory of simply connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X; \mathbb{Q})}(\mathbb{Q}, C^{\ast}(X; \mathbb{Q}))$ with a graded commutative algebra structure. Thus, we introduce new algebraic invariants referred to as the $Ext$-versions of the ordinary higher, module, and homology Topological Complexities of $X_0$, the rationalization of $X$. For Gorenstein spaces, we establish, under additional hypotheses, that the new homology topological complexity, denoted $HTC^{\mathcal{E}xt}_n(X,\mathbb{Q})$, lowers the ordinary $HTC_n(X)$ and, in case of equality, we extend Carasquel's characterization for $HTC_n(X)$ to some class of Gorenstein spaces (Theorem 1.2). We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristic especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.
References
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Year 2025,
Volume: 54 Issue: 2, 352 - 367, 28.04.2025
Abstract
References
- [1] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30, 305-330, 1955.
- [2] D. Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc. 2, 417-453, 1989.
- [3] R. Bott and H. Samelson, On the Pontryagin product in spaces of paths, Comment. Math. Helv. 27 (1), 320-337, 1953.
- [4] J. G. Carrasquel-Vera, Computations in rational sectional category, Bull. Belg. Math. Soc. Simon Stevin 22 (3), 455-469, 2015.
- [5] J. G. Carrasquel, Rational methods applied to sectional category and topological complexity, Contemp. Math. 702, 2018.
- [6] M. Farber, Topological Complexity of Motion Planning, Discrete Comput. Geom. 29, 211-221, 2003.
- [7] Y. Félix and S. Halperin, A note on Gorenstein spaces, J. Pure Appl. Algebra 223, 4937-4953, 2019.
- [8] Y. Félix and S. Halperin, Rational LS-category and its applications, Trans. Am. Math. Soc. 273 (1), 1-37, 1982.
- [9] Y. Félix, S. Halperin, C. Jacobson, C. Löfwall and J. C. Thomas, The radical of the homotopy Lie algebra, Amer. J. Math. 110, 301-322, 1988.
- [10] Y. Félix, S Halperin and J. C. Thomas, LS-catégorie et suite spectrale de Milnor- Moore, Bull. Soc. Math. France 111, 89-96, 1983.
- [11] Y. Félix, S. Halperin and J. C. Thomas, Gorenstein spaces. Adv. Math. 71, 92-112, 1988.
- [12] Y. Félix, S. Halperin and J. C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 215, Springer Verlag, 2000.
- [13] H. Gammelin, Gorenstein spaces with nonzero evaluation map, Trans. Amer. Math. Soc. 351 (8), 3433-3440, 1999.
- [14] S. Halperin, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra 83, 237-282, 1992.
- [15] S. Hamoun, Y. Rami and L. Vandembroucq, On the rational topological complexity of coformal elliptic spaces, J. Pure Appl. Algebra 227 (7), 107318, 2023.
- [16] L. Lechuga and A. Murillo, Complexity in rational homotopy, Topology 39, 89-94, 2000.
- [17] L. Lusternik and L. Shnirelmann, Méthodes Topologiques dans les problèmes variationnels, Hermann, Paris, 1934.
- [18] A. Murillo, The evaluation map of some Gorenstein algebras, J. Pure. Appl. Algebra 91, 209-218, 1994.
- [19] Y. B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157, 916- 920, 2010.
- [20] M. Spivak, Spaces satisfying Poincaré duality, Topology 6, 77-102, 1967.
There are 20 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | April 14, 2024 |
| Publication Date | April 28, 2025 |
| Published in Issue | Year 2025 Volume: 54 Issue: 2 |