Differentiable periodic maps
Differentiable periodic maps
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DOI：
10.1090/s0002-9904-1962-10730-7
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发表时间：
1964
期刊：
影响因子：
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通讯作者：
P. E. Conner;E. Floyd
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文献类型：
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作者：
P. E. Conner;E. Floyd
1. 边界主义团体。本文概述了作者将 Thorn 的配边理论 [ó] 应用到可微周期图研究中的努力。然而，首先我们将概述计算空间 [ l ] 的有向边群的方案。这些初步评论涉及 Milnor 提出的问题[4]。有限流形是具有边界的紧连通流形的有限不相交并，每个流形都带有 O 微分结构。有限流形 B 的边界用 dB 表示。闭流形是具有空边界的有限流形。我们现在定义一对 (X, ^4) 的有向边群。 (X, A) 中的有向奇异流形是有向有限流形的映射 f: (B} dB ) —»(X, A)。当且仅当存在有限定向流形 W 和映射 F: W—->X 时，这样的奇异流形在 (X, A) 中存在，使得 BC.dW 作为有限正则子流形，其方向由 W 的方向诱导，并且使得 F\ jB=/, F(dW— B) C.A.从两个这样的定向奇异流形 (Bl fx) 和 (£?, /2) 形成不相交联合 (B\\JB n 2l fxKJf2)，其中 B\C\B% = 0 和 / i U / 2 | £?==f,, * = 1 , 2。显然 ( £ » , f ) = ( J 3 n , f) 。我们认为两个奇异流形 (5J, /i) 和 (J5J,/2) 在 (X, yl) 中是相连的当且仅当不相交并集 (JB*U -~B1,f\\Jf(X, ^4) 和任何闭向流形 V 的模积由 [B, / ] [ F W ] = [BX V, g] 给出，其中 g(x9 y) =f(*)。对于任何映射 : (X, A)-*(Y, B) 存在诱导同态 f]。还有 d*: Qn(X, A)-*Qn-i(A)，由 d*([5», f ] ) = [3B», f\dB-*A] 给出。实际上 0*: &*(X, i4)-*Q*(F, 5 ) 和 d*: J2*(X, ^4)~>fts|c(^4) 是 0 和 1 度的 fl 模同态。
1. The bordism groups. This note presents an outline of the authors' efforts to apply Thorn's cobordism theory [ó] to the study of differentiable periodic maps. First, however, we shall outline our scheme for computing the oriented bordism groups of a space [ l ] . These preliminary remarks bear on a problem raised by Milnor [4]. A finite manifold is the finite disjoint union of compact connected manifolds with boundary each of which carries a O-differential structure. The boundary of a finite manifold, B, is denoted by dB. A closed manifold is a finite manifold with void boundary. We now define the oriented bordism groups of a pair (X, ^4). An oriented singular manifold in (X, A) is a map ƒ: (B} dB ) —»(X, A) of an oriented finite manifold. Such a singular manifold bords in (X, A) if and only if there is a finite oriented manifold W and a map F: W—->X such that BC.dW as a finite regular submanifold whose orientation is induced by that of W and such that F\ jB=/, F(dW— B) C.A. From two such oriented singular manifolds (Bl fx) and (£?, /2) a disjoint union (B\\JB n 2l fxKJf2) is formed with B\C\B% = 0 and / i U / 2 | £?==ƒ,, * = 1 , 2. Obviously ( £ » , ƒ ) = ( J 3 n , ƒ). We £ay that two singular manifold (5J, /i) and (J5J,/2) are bordant in (X, yl) if and only if the disjoint union (JB*U -~B1,f\\Jf(X, ^4) and any closed oriented manifold V the module product is given by [B, / ] [ F W ] = [BX V, g] where g(x9 y) =ƒ(*). For any map : (X, A)-*(Y, B) there is an induced homomorphism f]. There is also d*: Qn(X, A)-*Qn-i(A) given by d*([5», ƒ ] ) = [3B», f\dB-*A]. Actually 0*: &*(X, i4)-*Q*(F, 5 ) and d*: J2*(X, ^4)~>fts|c(^4) are fl-module homomorphisms of degree 0 and 1 .