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: http://higeom.math.msu.su/people/taras/cobordisms-en.pdf
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Дата индексирования: Sun Apr 10 00:41:06 2016
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COBORDISMS AND FINITE GROUP ACTIONS IN THE WORK OF TARAS PANOV In works of T. Panov [5][7] the cobordism theory and formal group laws are applied to the study of the fixed point sets of Z/p-actions. Characteristic classes and cob ordisms of manifolds with simple Z/p-action. A Z/p-action on a manifold M is called simple if all connected components of the fixed point set have trivial normal bundle. If additionally the tangent representations at the fixed points are same for all fixed submanifolds of the same dimension, then the action is strictly simple. In 1964 Conner and Floyd [2] proved that a cobordism class contains a manifold with a strictly simple Z/p-action if and only if all its characteristic numbers are zero mod p. S. P. Novikov in his pioneering work [3] of 1967 brought the methods of formal group laws into cobordism theory. This lead to a new approach to cobordism classification problems related to Z/p-actions. In particular, a significant breakthrough in the problem of describing the cobordism ring ideal generated by manifolds with a simple Z/p-action was achieved in [1]. However, the problem itself had remained unsolved until 1998, when T. Panov obtained its effective solution both in terms of the coefficients of the universal formal group law of geometric cobordisms and in terms of divisibility of characteristic numbers. Hirzebruch genera of manifolds with Z/p-action. The theme of cobordisms of Z/p-actions was further developed in works of Panov [5], [7], where he obtained formulae calculating most important Hirzebruch genera of manifolds with simple Z/p-action in terms of the invariants of the action. Hirzebruch genera, such as the Todd genus, signature or elliptic genus, constitute an important family of multiplicative cobordism invariants of manifolds. The problem of calculating these genera for manifolds with Z/p-action was posed in Buchstaber and Novikov's work [1], where the answer was obtained for the Todd genus. In Panov's work a general formula was obtained, expressing an arbitrary genus in terms of the weights of the tangent representations at the fixed points and the cobordisms classes of the fixed submanifolds. Most important particular cases were detailedly ^ treated. These include; the signature, the y -genus, the A-genus, the elliptic genus, and some other genera arising as the indices of complexes of elliptic operators. In his calculations, T. Panov used two different approaches; one due to BuchstaberNovikov, which reduces the calculation of a genus to calculating a certain number-theoretical trace of field extension, and another based on the application of the AtiyahSinger index theorem. The comparison of the formulae obtained by these two different methods led to non-trivial identities for Legendre and Chebyshev polynomials. The latter arise as the coefficients of the logarithm of the corresponding formal group laws. The results of these works of T. Panov constitute his PhD thesis, finished in 1999. The methods and ideas have since been implemented in the subsequent work of Panov on toric topology.