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Algebras of cellular cochains, and torus actions. (English. Russian original) Zbl 1068.52020

Russ. Math. Surv. 59, No. 3, 562-563 (2004); translation from Usp. Mat. Nauk 59, No. 3, 159-160 (2004).
From the text: We give a proof of the isomorphism of the integer cohomology algebra of the moment-angle complex \({\mathcal Z}_K\) [V. M. Bukhshtaber and T. E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series. 24, Providence, RI (AMS) (2002; Zbl 1012.52021)] and the Tor-algebra of the face ring of a simplicial complex \(K\). It is based on the construction of a cellular approximation of the diagonal map \(\Delta:{\mathcal Z}_K\to{\mathcal Z}_K\times{\mathcal Z}_K\). In cellular cochains there is no functorial associative multiplication, since in the general case one cannot choose a corresponding cellular approximation of the diagonal. The construction of the moment-angle complex is a functor from the category of simplicial complexes into the category of spaces with a torus action. We demonstrate that in the special case in hand the proposed cellular approximation of the diagonal is associative and functorial with respect to maps of moment-angle complexes induced by simplicial maps.

MSC:

52B70 Polyhedral manifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
57Q15 Triangulating manifolds

Citations:

Zbl 1012.52021
Full Text: DOI