上同伦不变量与泛上同伦不变量跳跃公式
Cohomotopy invariants and the universal cohomotopy invariant jump formula
摘要：
从Furuta的思想出发，我们给出了Hilbert丛之间的一类$S^1$ -等变非线性映射的上同伦不变量的一般构造方法.应用于Seiberg-Witten映射，这种形式主义产生了一类新的上同伦Seiberg-Witten不变量，它们关于4-流形的非同态具有明确的函子性质。对于具有$b_1=0$的四维流形，我们的不变量和Bauer-Furuta类是直接可比的;当$b_1=0$和$b_+>1$时它们是等价的，但在$b_1=0$，$b_+=1$的情况下更精细（它们检测跨壁现象）。我们在一个非常一般的框架中研究新的不变量的基本性质。特别地，我们证明了一个泛余同伦不变的跳跃公式和一个乘法性质。该形式主义适用于其他规范理论问题，例如规范理论（汉密尔顿）Gromov-Witten不变量理论。
英文摘要：
Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphisms of 4-manifolds. Our invariants and the Bauer-Furuta classes are directly comparable for 4-manifolds with $b_1=0$; they are equivalent when $b_1=0$ and $b_+>1$, but are finer in the case $b_1=0$, $b_+=1$ (they detect the wall-crossing phenomena). We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants.
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