环空间上的庞加莱对偶
Poincaré duality for loop spaces
AI总结：
本文研究了Rabinowitz Floer同调与上同调的结构，证明了其作为带权Frobenius代数的庞加莱对偶性，并将其推广到带权开-闭TQFT之间。
摘要：
我们证明了Rabinowitz Floer同调与上同调具有带权Frobenius代数的结构，对于闭合和开放弦均成立。我们证明了同调与上同调之间的庞加莱对偶定理，保持此结构。这提升为带权开-闭TQFT之间的对偶定理。我们系统地使用Tate向量空间的正式主义。在cotangent丛的情况下，我们定义Rabinowitz循环同调与上同调，并从统一视角解释了多年来在寻找闭合测地线过程中观察到的双结果。这些涉及临界水平、与基于循环空间的关系、所有测地线均为闭合的流形、Bott指数迭代、以及水平-幂性。此外，带权Frobenius代数结构为Sullivan所推测的循环乘积与余乘积之间的关系提供了意义和证明。
英文摘要：
We show that Rabinowitz Floer homology and cohomology carry the structure of a graded Frobenius algebra for both closed and open strings. We prove a Poincaré duality theorem between homology and cohomology that preserves this structure. This lifts to a duality theorem between graded open-closed TQFTs. We use in a systematic way the formalism of Tate vector spaces. Specializing to the case of cotangent bundles, we define Rabinowitz loop homology and cohomology and explain from a unified perspective pairs of dual results that have been observed over the years in the context of the search for closed geodesics. These concern critical levels, relations to the based loop space, manifolds all of whose geodesics are closed, Bott index iteration, and level-potency. Moreover, the graded Frobenius algebra structure gives meaning and proof to a relation conjectured by Sullivan between the loop product and coproduct.
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