Topological complexity and Lusternik Schnirelmann category of manifolds

Subject Areas : Statistics

Fezzeh Akhtarifar ¹ , Mohammad Ali Asadi Golmankhaneh ²

1 - Department of Applied Mathematics, Faculty of Basic Science, Urmia University, Urmia, Iran.
2 - Department of Applied Mathematics, Faculty of Basic Science, Urmia University, Urmia, Iran.

Received: 2019-07-17 Accepted : 2021-05-04 Published : 2021-05-22

Keywords: طول ناوی, رسته ی قوی, ضرب گوه ای, طول مخروطی,

Abstract :

Lusternik schnirelmann category and topological complexity are important invariant of topological spaces, now a days a lot of mathematician are interested to work in this area. In this paper in order to detect properties of spaces, we will compute Lusternik schnirelmann category and topological complexity of some of these spaces by computing the cup-length and zero-cup-length. These include the manifolds that we will calculate for the topological complexity and Lusternik Schnirelmann category are some of the Gressmannian manifolds, such as G_2 (R^4), and the products of manifolds, especially the products of the real projective spaces and their wedge products. Let TC(X) denotes the topological complexity of the path connected topological space X, and also cat(X) denots the Lusternik Schnirelmann category of topological space X. In the calculation of these numbers, we will first compute the upper and lower bounds of these invariants for considerable spaces, and we will try to approximate the boundaries with the methods and techniques to get the exact number. In this paper, we will use the cup-length and zero divisors cup length of spaces as the lower bounds for calculating TC and cat that are important computational tools for calculating these numbers.

References:

[1] O. Cornea, G. Lupton, J. Oprea, and D. Tanre´,  Lusternik -Schnirelmann category، Mathematical Surveys and Monographsm vol. 103, American Mathematical Society، Providence, RI, 2003 MR 1990857 (2004E:5001)

 

[2] M. Farber، S. Tabachnikov and S. Yuzvinsky, Topological robotics: Motion planning in projective spaces، Preprint arxiv :math. AT/ 0210018, 2 Oct (2002)

 

[3] M. Farber, Topological complexity of motion planning, Discrete Comput. Geom. 29, (2003) no. 2, 211-221.

 

[4] M. Farber، Instabilities of robot motion Topology Appl. 140, (2004) no. 3-2, 245-266.

 

[5] M. Farber, Topology of robot motion planning Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem, vol.،217, Springer, Dordrecht, 2006, pp. 185-230.

 

[6] I. M. James, On category, in the sense of Lusternik -schnirelmann، Topology, 331-348.

 

[7] N. Iwase, M. Mimura, T. Nishimoto, Lusternik -Schnirelmann category of non-simply connected compact simple Lie groups، Topology Appl. 150 (2005),
111-123.

 

[8] N. Iwase, K. Kikuchi, M. Miyauchi, On the Lusternik - Schnirelmann category of SO (10), March 21, 2019.

 

[9] H. L. Hiller, On the Cohomology of Real Grassmanians, Transactions of the American Mathematical Society. vol.257 No. 2Feb, (1980) pp. 521-533.

 

[10] C. A. I. Zapata، topological complexity of wedges, arXiv: 0677971. 1712 [math. AT] 19 Dec 2017.