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Recently, mathematical analysis clarified that sp;{2} hybridized carbon should have a three-dimensional crystal structure (K4) which can be regarded as a twin of the sp;{3} diamond crystal. In this study, various physical properties of the K4 carbon crystal, especially for the electronic properties, were evaluated by first principles calculations. Although the K4 crystal is in a metastable state, a possible pressure induced structural phase transition from graphite to K4 was suggested. Twisted pi states across the Fermi level result in metallic properties in a new carbon crystal.

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L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CLOSED ORBITS IN HOMOLOGY CLASSES by ATSUSHI KATSUDA and TOSHIKAZU SUNADA Dedicated to Pr. Akihiko Morimoto for his 60th birthday 6 ATSUSHI KATSUDA AND TOSHIKAZU SUNADA hence the winding cycle is regarded as the average of the c< homological " direction in which the orbits are traveling. The central limit theorem (cf. Denker and Philipp [5]) guarantees the existence of the limit 8( (9^ x) rfr-^O(co)) 2 , v •/x \Jo / which yields a positive semi-definite quadratic form on H^X, R). We call 8 the cornnance form. As we will see later, 8 is positive definite on Ker 0, and hence gives rise to a Euclidean metric on Ker 0. Consider the character group H ofH. The tangent space T^ H at the trivial character 1 is identified with the dual H 1 ' == Hom(H,R), which is also identified, in a natural manner, with a subspace in H^X,^. Therefore if 0 vanishes on H 1 ', the covariance form induces a flat metric on the group H. We denote by vol(H) the volume with respect to the metric. Theorem 1 (Density theorem).-If 0 vanishes on the dual H 1 ', then (0.1) 7r(^a)-G^^ asx^co, where b == rank H and C == {2^)-^ vol(H)-1 h~\ The above condition on the winding cycle is necessary for the asymptotic like (0.1). In fact we have the following Theorem 2.-7/'