Infinite loop spaces and positive scalar curvature
Infinite loop spaces and positive scalar curvature
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DOI：
10.1007/s00222-017-0719-3
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发表时间：
2014-11
影响因子：
3.1
通讯作者：
B. Botvinnik;Johannes Ebert;O. Randal-Williams
中科院分区：
文献类型：
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作者：
B. Botvinnik;Johannes Ebert;O. Randal-Williams
研究了高维紧致自旋流形上正数量曲率度量空间的同伦型。希钦利用正标量曲率流形上不存在调和旋量的事实，构造了从正标量度量空间到实K-理论谱中一个合适空间的次级指数映射。我们的主要结果是关于这个映射的非平凡性。本文证明了对于，从Madsen-Tillmann-韦斯谱因子的无限圈空间（直到同伦）通过任意2n维自旋流形上的正标量曲率度量空间的自然KO定向。对于奇维流形，我们证明了一个类似的因子分解的存在性。当与同伦理论的计算方法相结合时，这些结果具有很强的意义。例如，次指数映射在所有有理同伦群上都是满射的。我们还提出了更精细的计算积分同伦群。为了证明我们的结果，我们使用了三套主要的技术工具和结果。第一套工具来自黎曼几何：我们使用参数化版本的Gromov-Lawson手术技术，使我们能够应用同伦理论技术的度量空间的正标量曲率。其次，我们将Hitchin的二级指标与其他几个指标理论的结果，如Atiyah-Singer族指标定理，非紧流形上的指标的可加性定理和谱流指标定理。最后，我们使用的结果和工具，最近在研究模空间的流形和配边范畴。我们在本文中使用的关键新成分是由Galatius和第三名作者证明的Madsen-Weiss定理的高维模拟。
We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the realK-theory spectrum. Our main results concern the nontriviality of this map. We prove that for, the naturalKO-orientation from the infinite loop space of the Madsen–Tillmann–Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any 2n-dimensional spin manifold. For manifolds of odd dimension, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov–Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin’s secondary index to several other index-theoretical results, such as the Atiyah–Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen–Weiss theorem, proven by Galatius and the third named author.