An algebraic classification of noncompact 2-manifolds
1971; American Mathematical Society; Volume: 156; Linguagem: Inglês
10.1090/s0002-9947-1971-0275436-9
ISSN1088-6850
Autores Tópico(s)Advanced Differential Equations and Dynamical Systems
ResumoIntroduction.We prove the following: Theorem.A surface S is determined, up to homeomorphism, if the following are known:(1) the vector space H}(S, Z2),(2) the Boolean ring H°(S, Z2),(3) the cup products u : HXS, Z2) ® Hl(S, Z2) -► H2(S, Z2), u : H\S, Z2) ® HÎ(S, Z2) -> H?(S, Z2), u : H°(S, Z2) ® H\S, Z2) -> H¡(S, Z2).By a surface we mean a connected separable 2-manifold without border.Hq(S, Z2) and Hqc(S, Z2) denote, respectively, the standard and compact support cohomology groups.H §(S, Z2) is the cohomology group of the cochains of S modulo the cochains of S with compact support (see §1).For a closed surface the Theorem is a direct consequence of the Classification Theorem for Closed Surfaces.In this case H°(S, Z2) = 0.For an open surface the Theorem is an algebraic version of the Kerekjarto Classification Theorem ([9, p. 262] or [5, Chapter 5]).The Kerekjarto Theorem says that an open surface is topologically determined, modulo a compact subsurface, by its space of ends e(S) [1, p. 82] and various subspaces of e(S).In this paper we substitute for e(S) the Boolean ring H°(S, Z2).In [2] we proved the Theorem by using (1) the Kerekjarto Theorem, (2) the Stone Representation Theorem [4, p. 168] which says that the compact totally disconnected space e(S) is determined by the Boolean ring of continuous functions from e(S) to Z2, and (3) a result shown to me by W. S. Massey which says that this ring is isomorphic to H°(S, Z2) (see Remark 1.7).In this paper the Theorem is proven directly, without explicitly using ends.Essentially we follow the proof of the Kerekjarto Theorem as presented in [9, §4], changing the statements about e(S) to statements about H°(S, Z2).The above theorem occurs as Corollary 4.2 to Theorem 4.1, which is a more specific result.
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