Far Eastern Mathematical Journal

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Pontryagin algebras of some moment-angle-complexes

Veryovkin Y.A.

2016, issue 1, Ñ. 9-23

We consider the problem of describing the Pontryagin algebra (loop homology) of moment-angle complexes and manifolds. The moment-angle complex $Z_K$ is a cell complex built of products of polydiscs and tori parametrised by simplices in a finite simplicial complex K. It has a natural torus action and plays an important role in toric topology. In the case when K is a triangulation of a sphere, $Z_K$ is a topological manifold, which has interesting geometric structures. Generators of the Pontryagin algebra $H_*(\Omega Z_K)$ when K is a flag complex have been described in the work of Grbic, Panov, Theriault and Wu. Describing relations is often a difficult problem, even when K has a few vertices. Here we describe these relations in the case when K is the boundary of a pentagon or a hexagon. In this case, it is known that $Z_K$ is a connected sum of products of spheres with two spheres in each product. Therefore $H_*(\Omega Z_K)$ is a one-relator algebra and we describe this one relation explicitly, therefore giving a new homotopy-theoretical proof of McGavran's result. An interesting feature of our relation is that it includes iterated Whitehead products which vanish under the Hurewicz homomorphism. Therefore, the form of this relation cannot be deduced solely from the result of McGavran.

moment-angle complex, Pontryagin algebra

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[1] V. M. Buchstaber and T. E. Panov, «Toric Topology», Math. Surv. and Monogr., Amer. Math. Soc., 204 (2015).
[2] J. Grbic, T. Panov, S. Theriault and J. Wu, «Homotopy types of moment-angle complexes for ag complexes», Trans. Amer. Math. Soc., 2015, arXiv: 1211.0873.
[3] D. McGavran, «Adjacent connected sums and torus actions», Trans. Amer. Math. Soc., 251 (1979), 235–254.
[4] S. Gitler and S. Lopez de Medrano, Intersections of quadrics, moment-angle manifolds and connected sums, Preprint, 2009, arXiv: 0901.2580.
[5] F. Bosio and L. Meersseman, «Real quadrics in Cn, complex manifolds and convex polytopes», Acta Math., 197:1 (2006), 53–127.
[6] Wolfram Mathematica, A software system devoted to supporting research in mathematica. Avaliable at http://www.wolframalpha.com/.
[7] A. Hatcher, Algebraic topology, MCCME, Moscow, 2011.
[8] Macaulay2, A software system devoted to supporting research in algebraic geometry and commutative algebra. Avaliable at http://www.math.uiuc.edu/Macaulay2/.

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