高连通流形的映射类群的阿贝尔商
Abelian quotients of mapping class groups of highly connected manifolds
摘要：
我们计算了流形$W_g^{2n} = g(S^n \times S^n)$的映射类群对$n \geq 3$和$g \geq 5$的交换.答案是两部分的直和。第一部分由映射类群对中同调的作用产生，并在中同调的自同构群的阿贝尔化中取值。第二部分产生于bordism类的映射柱和价值观的商的稳定同伦群的领域由一定的子群，在许多情况下同意的形象稳定的$J$ -同态。我们将其计算与一个纯粹的同伦理论问题。
英文摘要：
We compute the abelianisations of the mapping class groups of the manifolds $W_g^{2n} = g(S^n \times S^n)$ for $n \geq 3$ and $g \geq 5$. The answer is a direct sum of two parts. The first part arises from the action of the mapping class group on the middle homology, and takes values in the abelianisation of the automorphism group of the middle homology. The second part arises from bordism classes of mapping cylinders and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable $J$-homomorphism. We relate its calculation to a purely homotopy theoretic problem.
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