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首页> 外文期刊>Canadian Journal of Mathematics >On Homotopy Invariants of Combings of Three-manifolds
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On Homotopy Invariants of Combings of Three-manifolds

机译：关于三流形梳的同伦不变量

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Combings of compact, oriented, 3-dimensional manifolds M are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying Spine-structure. A combing is called torsion if this Euler class is a torsion element of H-2 (M; Z). Gompf introduced a Q-valued invariant theta(G) of torsion combings on closed 3-manifolds, and he showed that theta(G) distinguishes all torsion combings with the same Spin(c)-structure. We give an alternative definition for theta(G) and we express its variation as a linking number. We define a similar invariant p(1) of combings for manifolds bounded by S-2. We relate p(1) to the Theta-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formulae Theta = 1/4p(1) + 6 lambda((M) over cap), where lambda is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.
机译：紧凑，定向的3维流形M的梳是无处消失的矢量场的同伦类。正常束的Euler类是精梳的不变性，它仅取决于基础的Spine结构。如果此Euler类是H-2（M; Z）的扭转元素，则该梳理称为扭转。 Gompf在闭合的3个流形上引入了Q值不变的扭梳theta（G），他证明了theta（G）可以区分所有具有相同Spin（c）结构的扭梳。我们给出theta（G）的替代定义，并将其变化形式表示为链接数。我们为以S-2为边界的流形定义了相似的梳理不变p（1）。通过公式Theta = 1 / 4p（1）+ 6 lambda（（M）over cap），我们将p（1）与Theta不变量相关联，它是有理同源性3球的最简单的配置空间积分不变量，其中lambda是Casson-Walker不变式。文章还包括3个流形的梳理的独立显示。

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