定号4-流形的同义反复类
Tautological classes of definite 4-manifolds
AI总结：
本文证明了紧致光滑单连通定号4-流形的同义反复类对角化定理，利用Seiberg-Witten理论的族版本方法，完全确定了$\mathbb{CP}^2$等流形的同义反复环并推导出普遍成立的线性关系。
摘要：
我们证明了紧致、光滑、单连通定号4-流形的同义反复类（或称广义Miller-Morita-Mumford类）的对角化定理。我们的结果可视为Donaldson对角化定理的族版本。我们使用Seiberg-Witten理论的Bauer-Furuta上同伦加细的族版本来证明该结果。我们利用主结果推导出关于此类4-流形的同义反复类的各种结论。特别地，我们完全确定了$\mathbb{CP}^2$和$\mathbb{CP}^2 \\# \mathbb{CP}^2$的同义反复环。我们还推导出同义反复环中的一系列线性关系，这些关系在普遍意义下成立，即它们对所有紧致、光滑、单连通定号4-流形均成立。
英文摘要：
We prove a diagonalisation theorem for the tautological, or generalised Miller-Morita-Mumford classes of compact, smooth, simply-connected definite $4$-manifolds. Our result can be thought of as a families version of Donaldson's diagonalisation theorem. We prove our result using a families version of the Bauer-Furuta cohomotopy refinement of Seiberg-Witten theory. We use our main result to deduce various results concerning the tautological classes of such $4$-manifolds. In particular, we completely determine the tautological rings of $\mathbb{CP}^2$ and $\mathbb{CP}^2 \# \mathbb{CP}^2$. We also derive a series of linear relations in the tautological ring which are universal in the sense that they hold for all compact, smooth, simply-connected definite $4$-manifolds.
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