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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Riemann–Roch theorem for smooth manifolds ： ウィキペディア英語版 Riemann–Roch theorem for smooth manifolds In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure. ==Formulation== Let ''X'' and ''Y'' be oriented smooth closed manifolds, and ''f'': ''X'' → ''Y'' a continuous map. Let ''v''''f''=''f'' *(''TY'') − ''TX'' in the K-group K(X). If dim(X) ≡ dim(Y) mod 2, then :$\backslash mathrm(f\_(x))\; =\; f\_(\backslash mathrm(x)\; e^\backslash hat(v\_f)),$ where ch is the Chern character, d(vf) an element of the integral cohomology group ''H''2(''Y'', ''Z'') satisfying ''d''(''v''''f'') ≡ ''f'' * ''w''2(T''Y'')-''w''2(T''X'') mod 2, fK * the Gysin homomorphism for K-theory, and fH * the Gysin homomorphism for cohomology .〔M. Karoubi, ''K-theory, an introduction'', Springer-Verlag, Berlin (1978)〕 This theorem was first proven by Atiyah and Hirzebruch.〔M. Atiyah and F. Hirzebruch, ''Riemann–Roch theorems for differentiable manifolds'' (Bull. Amer. Math. Soc. 65 (1959) 276–281)〕 The theorem is proven by considering several special cases.〔M. Karoubi, ''K-theory, an introduction'', Springer-Verlag, Berlin (1978)〕 If ''Y'' is the Thom space of a vector bundle ''V'' over ''X'', then the Gysin maps are just the Thom isomorphism. Then, using the splitting principle, it suffices to check the theorem via explicit computation for line bundles. If ''f'': ''X'' → ''Y'' is an embedding, then the Thom space of the normal bundle of ''X'' in ''Y'' can be viewed as a tubular neighborhood of ''X'' in ''Y'', and excision gives a map :$u:H^$ *(B(N), S(N)) \to H^ *(Y, Y-B(N)) \to H^ *(Y) and :$v:K(B(N),\; S(N))\; \backslash to\; K(Y,\; Y-B(N))\; \backslash to\; K(Y)$. The Gysin map for K-theory/cohomology is defined to be the composition of the Thom isomorphism with these maps. Since the theorem holds for the map from ''X'' to the Thom space of ''N'', and since the Chern character commutes with ''u'' and ''v'', the theorem is also true for embeddings. ''f'': ''X'' → ''Y''. Finally, we can factor a general map ''f'': ''X'' → ''Y'' into an embedding :$i:\; X\; \backslash to\; Y\; \backslash times\; S^$ and the projection :$p:\; Y\; \backslash times\; S^\; \backslash to\; Y.$ The theorem is true for the embedding. The Gysin map for the projection is the Bott-periodicity isomorphism, which commutes with the Chern character, so the theorem holds in this general case also. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemann–Roch theorem for smooth manifolds」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.