Symplectic homology and the Eilenberg-Steenrod axioms
The authors say that there are too many Floer (co)homologies for noncompact objects, e.g., wrapped Floer homology, symplectic homology, Rabinowitz-Floer homology and linearized contact homology. In this paper, based on the notion Liouville cobordism with filling, six types of symplectic homologies are introduced in a uniform way (\S2). They are shown to satisfy the Eilenberg-Steenrod axioms except the dimension axiom (Theorem 1.4. proved in \S3--\S7), and the Rabinowitz-Floer homology and the wrapped Floer homology are derived from them (\S8). A Liouville domain is an exact symplectic manifold \((W^{2n},\omega,\lambda)\), \(\omega=d\lambda\) such that \(\alpha=\lambda|_{\partial W}\) is a positive contact form. A Liouville cobordism is defined similarly, but so that \(\alpha\wedge(d\alpha)^{n-1}\) defines the opposite of the boundary orientation on some of components of \(\partial W\), which is the negative boundary \(\partial^-W\) of \(W\). The positive boundary \(\partial^+W\) of \(W\) is defined by \(\partial W\setminus\partial^-W\). A filling \(F\) of the Liouville cobordism is a Liouville domain whose positive boundary is isomorphic to the negative boundary of \(W\) so that the concatenation \(F\circ W\) is a Liouville domain with positive boundary \(\partial^+W\). On a Liouville cobordism with filling \(F\) the following (smooth time-dependent 1-periodic approximations of a) Hamiltonian \(H_{\mu,\tau}\), such that \[ \begin{aligned} H_{\mu.\tau}=&\tau r-\tau \;\mathrm{on} \;[1,\infty)\times \partial^+W, \quad H_{\mu,\tau}=0 \;\mathrm{on} \;W, \\ H_{\mu,\tau}=&-\mu r+\mu \mathrm{on} \;(\delta,1]\times\partial^-W\subset F,\end{aligned} \] and \(H_{\mu,\tau}\) is constant on the remaining part of \(F\) is constructed. The Hamiltonian Floer homology defined by \(H_{\mu,\tau}\), truncated in the finite action window \((a,b)\) is denoted by \(\mathrm{HF}_\ast^{(a,b)}(H_{\mu,\tau})\). Taking limits with respect to \((a,b)\), six variants of symplectic homologies of a cobordism, the action-filtered symplectic homologies, \(\mathrm{SH}_\ast^\heartsuit(W)\), \(\heartsuit\in\{\emptyset, >0,\geq 0, =0, \leq 0, <0\}\) are defined (Definition 2.8). Especially, \(\mathrm{SH}^\emptyset_\ast(W)\) is the symplectic homology \(\mathrm{SH}_\ast(W)\). A pair of Liouville cobordisms \((W,V)\) consists of a Liouville cobordism \((W,\omega,\lambda)\) together with a codimension-0 submanifold with boundary \(V\) such that \((V,\omega|_V,\lambda|_V)\) is a Liouville cobordism and \((W\setminus V, \omega,\lambda)\) is the disjoint union of two Liouville cobordisms \(W^{\mathrm{bottom}}\) and \(W^{\mathrm{top}}\) such that \(W=W^{\mathrm{bottom}}\circ V\circ W^{\mathrm{top}}\). Here, \(W^{\mathrm{bottom}}, W^{\mathrm{top}}\) and \(V\) allowed to be empty. For a Liouville cobordism pair \((W,V)\) with filling \(F\), relative symplectic homologies \(\mathrm{SH}^\heartsuit_\ast(W.V)\) are defined similar to \(\mathrm{SH}^\heartsuit_\ast(W)\). Then this paper proves the following. Theorem 1.4. The action-filtered symplectic homology group \(\mathrm{SH}^\heartsuit_\ast\) with coefficients in a field \(\mathbb{K}\) defines a Liouville homology theory. Here, a Liouville homology theory is a contravariant functor \((W,V)\to \mathbf{H}_\ast(W, V)\) from the Liouville category with fillings to the category of graded \(\mathbb{K}\)-vector spaces, having the Viterbo transfer map \[ f_!:\mathbf{H}_\ast(W^\prime,V^\prime)\to \mathbf{H}_\ast(W,V), \] where \(f:(W,V)\to (W^\prime,V^\prime)\) is an exact embedding between pairs with fillings, satisfying the following properties: {\parindent=0.8cm \begin{itemize}\item[1.] \textbf{Homotopy}. If \(f\) and \(g\) are homotopic through exact embeddings, then \(f_!=g_!\). \item[2.] \textbf{Exact triangle of a pair}. Given a pair \((W,V)\) for which we have the inclusions \(V\to W\to (W,V)\), there is a functorial exact triangle in which the map \(\partial\) has degree -1. \item[3.] \textbf{Excision}. For any cobordism triple \((W,V,U)\), the transfer map induced by the inclusions \((i:W\setminus\mathrm{Int}(U), V\setminus\mathrm{Int}(U))\to (W,V)\) is an isomorphism \[ i_!:\mathbf{H}_\ast(W,V)\cong\mathbf{H}_\ast(W\setminus\mathrm{Int}(U),V\setminus\mathrm{Int}(U)). \] \end{itemize}} (cf. Theorem 1.2. and Definition 1.3). The definitions of Liouville cobordism and symplectic homology for filled Liouville cobordism are stated in \S1. But details are given in \S2. In \S3, duality of homologies are discussed and symplectic cohomology for filled Liouville cobordisms \(\mathrm{SH}_\heartsuit^\ast(W,V)\), \(\heartsuit\in\{\emptyset, >0, \geq 0, =0, \leq 0, <0\}\) is defined (Definitions 3.2 and 3.1). If they have field coefficients, they satisfies \[ \begin{aligned} &\mathrm{SH}^k_\heartsuit(W,V)\cong \mathrm{SH}^\heartsuit_k(W,V)^\vee, \quad \heartsuit\in\{>0,\geq 0, =0\}, \\ &\mathrm{SH}^\heartsuit_k(W,V)\cong \mathrm{SH}^k_\heartsuit(W,V)^\vee, \quad \heartsuit\in\{<0,\leq 0, =0 \}, \end{aligned} \] see Theorem 3.5. Then, the Poincaré duality \[ \mathrm{SH}_\ast^\heartsuit(W,A)\cong \mathrm{SH}^{-\ast}_{-\heartsuit}(W, A^c) \] is proved in Theorem 3.4. Here \(A\subset\partial W\) is an admissible union of connected components, and \(-\heartsuit\) takes the values \(\emptyset,<0, \leq 0, =0, \geq 0, >0\) respectively. After studying the mapping cone \(C(f)\) from the point of view of symplectic homology in \S4, the transfer map \(f_!^\heartsuit:\mathrm{SH}^\heartsuit_\ast(W)\to \mathrm{SH}^\heartsuit_\ast(V)\) is defined in \S5 [\textit{C. Viterbo}, Geom. Funct. Anal. 9, No. 5, 985--1033 (1999; Zbl 0954.57015)], and its homotopy invariance is proved (Proposition 5.5). Excision is proved in \S6 (Theorem 6.8) by using the staircase Hamiltonian on \(\hat{W}_F\). Excision for triples \[ \mathrm{SH}^\heartsuit_\ast(W,V)\cong\mathrm{SH}^\heartsuit_\ast(\overline{W\setminus V}, \overline{V\setminus U}), \] where \((W,V,U)\) is a filled Liouville cobordism triple, is also shown (Theorem 6.11). Exact triangles of a pair of filled Liouville cobordisms both in homology (degree of boundary is -1) and cohomology (degree of coboundary is 1) are proved in \S7 (Theorem 7.1), via choosing cofinal families of Hamiltonians (cf. Figures 16, 17, 18). The Mayer-Vietoris exact triangle, when \(W=U\cup V\), \(Z=U\cap V\) and \(U=u^{\mathrm{bottom}}\circ Z\), \(V=Z\circ V^{\mathrm{top}}\), \(W=U^{\mathrm{bottom}}\circ Z\circ V^{\mathrm{top}}\) is also proved (Theorem 7.17). \S8 treats a variant of symplectic homology groups. Especially, Rabinowitz-Floer homology is shown to be the symplectic homology of the trivial cobordism over \(\partial V\), i.e., \(\mathrm{RFH}_\ast(\partial V)\cong\mathrm{SH}_\ast(\partial V)\). \(S^1\)-equivalent symplectic homologies and wrapped Floer homology (Lagrangian symplectic homology) are also treated. \S9 derives several already known results on Floer homology from the arguments of this paper. Some of them are the following: {\parindent=0.8cm \begin{itemize}\item[1.] Poincaré duality and exact triangle of \((V,\partial V)\): Theorem 9.5 and [\textit{K. Cieliebak} et al., Ann. Sci. Éc. Norm. Supér. (4) 43, No. 6, 957--1015 (2010; Zbl 1213.53105)]. \item[2.] Vanishing and finite dimensionality: Corollary 9.9 and Theorem 9.11 (displaceability implies vanishing) [\textit{K. Cieliebak} and \textit{U. A. Frauenfelder}, Pac. J. Math. 239, No. 2, 251--316 (2009; Zbl 1221.53112); \textit{J. Kang}, Geom. Dedicata 170, 135--142 (2014; Zbl 1300.53070)]. \item[3.] Invariants of contact manifolds: Proposition 9.17 and [\textit{F. Bourgeois} and \textit{A. Oancea}, Invent. Math. 175, No. 3, 611--680 (2009; Zbl 1167.53071)]. \item[4.] Subcritical handle attaching: Proposition 9.19 and [\textit{K. Cieliebak}, J. Eur. Math. Soc. (JEMS) 4, No. 2, 115--142 (2002; Zbl 1012.53066)]. \end{itemize}} This section concludes by the conjecture that the surgery exact triangle of \textit{F. Bourgeois} et al. [Geom. Topol. 16, No. 1, 301--389 (2012; Zbl 1322.53080)], can be reinterpreted in the language of this paper (Conjecture 9.21). In \S10, the last section, with applications to TQFT in mind [\textit{P. Seidel}, in: Current developments in mathematics, 2006. Somerville, MA: International Press. 211--253 (2008; Zbl 1165.57020)], the pair-of-pants product on Floer homology is shown to induce a product on \(\mathrm{SH}_\ast(W,A)\), etc., where \(A\) is an admissible union of boundary components. Properties of the product are also studied (Theorem 10.2). In the appendix, written with P. Albert, calling a Liouville cobordism SAWC if the unit \(1_W\) in \(H^0(W)\) is mapped to zero under the map \(H^0(W)\cong\mathrm{SH}^{=0}_n(H)\to \mathrm{SH}^{\geq 0}_n(W)\), \(W\) and \(\partial_-W\) are shown to be SAWC, if \(\partial_+W\) is SAWC, and the first Chern class of \(W\) vanishes, and \(\partial_-W\) is hypertight, index-positive or Liouville filliable (Proposition A.2). From this proposition, several non-existence results of Liouvile cobordism, such as non existence of Liouville cobordism \(W\) with \(\partial_-W\) hypertight and \(\mathrm{SH}_\ast(\partial_+W)=0\), etc., are derived.
Eilenberg-Steenrod axioms for a homology theory
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