Stable moduli spaces of high-dimensional manifolds
Stable moduli spaces of high-dimensional manifolds
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高维流形的稳定模空间
DOI：
10.1007/s11511-014-0112-7
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发表时间：
2012
期刊：
影响因子：
3.7
通讯作者：
O. Randal
中科院分区：
文献类型：
--
作者：
Søren Galatius;O. Randal
我们证明了高维流形的一个类似的Madsen-Weiss定理。特别地，我们明确地描述了光滑纤维束的特征类环，其纤维束是Sn×Sn的g个拷贝的连通和，其极限为$${g \to \infty}$$ g→∞。合理地说，它是在某些显式生成器中的多项式环，给出了芒福德猜想的高维类比。更一般地，我们研究了相对于P为（N - 1）连通的固定（2n-1）维流形P的零边的模空间$${\mathcal{N}(P)}$$ N(P)。我们利用P的某些自边确定了稳定后$${\mathcal{N}(P)}$$ N(P)的同调。稳定同调与无限环空间的同调一致。
We prove an analogue of the Madsen–Weiss theorem for high-dimensional manifolds. In particular, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of Sn×Sn, in the limit $${g \to \infty}$$g→∞. Rationally it is a polynomial ring in certain explicit generators, giving a high-dimensional analogue of Mumford’s conjecture.More generally, we study a moduli space $${\mathcal{N}(P)}$$N(P) of those null-bordisms of a fixed (2n–1)-dimensional manifold P which are (n–1)-connected relative to P. We determine the homology of $${\mathcal{N}(P)}$$N(P) after stabilisation using certain self-bordisms of P. The stable homology is identified with that of an infinite loop space.