The configuration space of the $$n$$-arms machine in the Euclidean space.(English)Zbl 1124.55004

The authors determine the configuration space of a specific spider-like robotic machine in $${\mathbb R}^d$$ for $$d\geq 3$$. A configuration of their machine is given by specifying $$n+1$$ points $$C, N_1, N_2, \dots, N_n\in {\mathbb R}^d$$ satisfying the following conditions
$| | N_i-C| | =| | N_i-B_i| | =r/2, \quad i=1, \dots, n.$
Here $$B_1, \dots, B_n$$ are fixed points situated at vertices of a regular planar $$n$$-gon, $B_i=(\cos \frac{2\pi i}{n}, \sin \frac{2\pi i}{n}, 0, \dots, 0), \quad i=1, \dots, n.$ Hence, the machine has its “body” at point $$C$$ and $$n$$ “legs” of length $$r$$, each having two parts with a “knee”$$N_i$$ in the middle. The endpoints of the legs are anchored at points $$B_i$$. The length $$r$$ is supposed to satisfy the inequalities $$1<r<2$$ for $$n$$ even and $$1<r<2\cos \frac{2\pi}{n}$$ for $$n$$ odd. The main result of the paper, Corollary C, states that the configuration space of this mechanism is a wedge of spheres; more specifically the authors prove that the configuration space is homeomorphic to $\Sigma^{d-2}\left(\left(\bigvee\limits_{i=2}^{n-2}X_i\right) \vee S^{n(d-2)+2}\right),$ where $$X_i$$ denotes the wedge of $$n\binom{n-2}{i-1} - \binom{n}{i}$$ copies of the sphere $$S^{i(d-2)+1}$$.

MSC:

 55R80 Discriminantal varieties and configuration spaces in algebraic topology 57N65 Algebraic topology of manifolds 55R55 Fiberings with singularities in algebraic topology 70B15 Kinematics of mechanisms and robots
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 [1] Atiyah, M. F.; Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A, 308, 523-615 (1982) · Zbl 0509.14014 [2] Bott, R., Nondegenerate critical manifolds, Ann. of Math., 60, 248-261 (1954) · Zbl 0058.09101 [3] Bousfield, A. K.; Kan, D. M., Homotopy Limits, Completions and Localizations, Lecture Notes in Math., vol. 304 (1972), Springer: Springer Berlin · Zbl 0259.55004 [4] Dror-Farjoun, E., Cellular inequalities, (The Čech Centennial. The Čech Centennial, Contemp. Math., vol. 181 (1995), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0885.55005 [5] Dror-Farjoun, E., Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Math., vol. 1622 (1996), Springer: Springer Berlin · Zbl 0842.55001 [6] Dugger, D.; Isaksen, D. C., Hypercovers in topology [7] Eldar, D., Maps and machines, Hebrew Univ., MSc project, in preparation [8] Hastings, H. M., Fibrations of compactly generated spaces, Michigan Math. J., 21, 243-251 (1974) [9] Hausmann, J.-C.; Knutson, A., The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble), 48, 1998, 281-321 (1974) · Zbl 0903.14019 [10] Hirschhorn, P. S., Model Categories and Their Localizations, Math. Surveys Monographs, vol. 99 (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1017.55001 [11] Hollender, J.; Vogt, R. M., Modules of topological spaces, applications to homotopy limits and $$E_\infty$$ structures, Arch. Math., 59, 115-129 (1992) · Zbl 0766.55006 [12] Hovey, M., Model Categories, Math. Surveys Monographs, vol. 63 (1999), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0909.55001 [13] Kamiyama, Y.; Tezuka, M., Topology and geometry of equilateral polygon linkages in the Euclidean plane, Quart. J. Math. Oxford (2), 50, 463-470 (1999) · Zbl 0961.57014 [14] Kamiyama, Y.; Tezuka, M.; Toma, T., Homology of the configuration spaces of quasi-equilateral polygon linkages, Trans. Amer. Math. Soc., 350, 4869-4896 (1998) · Zbl 0998.55009 [15] Kapovich, M.; Millson, J., On the moduli space of polygons in the Euclidean plane, J. Differential Geom., 42, 430-464 (1995) · Zbl 0854.51016 [17] Shvalb, N.; Shoham, M.; Blanc, D., The configuration space of arachnoid mechanisms, Forum Math., 17, 1033-1042 (2005) · Zbl 1223.70011
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