论文标题
莫尔斯和浮子同源性中的循环相关
Loop coproduct in Morse and Floer homology
论文作者
论文摘要
通过众所周知的Viterbo定理，封闭歧管的cotangent捆绑包的符合性同源性与其循环空间的同源性同构。在本文中，我们将这种同构的范围扩展到多个方向。首先，我们从莫尔斯（Morse）的理论上对{\ em rabinowitz循环同源性}进行了直接定义，并证明其产品与Rabinowitz Floer同源性上的裤子对产品一致。该证明使用刺穿的Annuli的压实模量空间。其次，我们证明，当仅限于{\ em阳性}的浮动同源性时，相对于恒定回路，循环空间同源性相对于恒定循环，viterbo同构相互交织在一起的次级裤子对偶发的各种结构与循环同源性共同构造。第三，我们介绍{\ em降低的循环同源性}，这是循环产物的规范还原和循环同源性共同构图的扩展的共同定义领域，该循环同源性共同构图共同定义了交换性同时共同共同的Unital Intital Indital Infitionsimsimaliitsimal Antimitaliitsimal抗合性的抗对称性Bialgebra。在此过程中，我们表明，通过使用线性汉密尔顿和能量函数的平方根，可以将abbondolo-schwarz的准同构从二次汉密尔顿人的浮子复合物转变为能量功能的莫尔斯复合物。
By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper we extend the scope of this isomorphism in several directions. First, we give a direct definition of {\em Rabinowitz loop homology} in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to {\em positive} Floer homology, resp.~loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce {\em reduced loop homology}, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo-Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism by using linear Hamiltonians and the square root of the energy functional.
