The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Publications mathématiques de l'IHÉS Pub Date : 2024-04-02 DOI:10.1007/s10240-024-00147-9

Vincent Colin, Paolo Ghiggini, Ko Honda

引用次数: 0

Abstract

Given a closed oriented 3-manifold \(M\), we establish an isomorphism between the Heegaard Floer homology group \(HF^{+} (-M)\) and the embedded contact homology group \(ECH(M)\). Starting from an open book decomposition \((S,\mathfrak{h} )\) of \(M\), we construct a chain map \(\Phi ^{+}\) from a Heegaard Floer chain complex associated to \((S,\mathfrak{h} )\) to an embedded contact homology chain complex for a contact form supported by \((S,\mathfrak{h} )\). The chain map \(\Phi ^{+}\) commutes up to homotopy with the \(U\)-maps defined on both sides and reduces to the quasi-isomorphism \(\Phi \) from (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024a, 2024b) on subcomplexes defining the hat versions. Algebraic considerations then imply that the map \(\Phi ^{+}\) is a quasi-isomorphism.

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Heegaard Floer 同调与嵌入接触同调的等价性 III：从帽子到加号

给定一个封闭的面向 3-manifold（M），我们在 Heegaard Floer 同调群（HF^{+} (-M)\) 和嵌入接触同调群（ECH(M)\) 之间建立了同构关系。从\(M)的开卷分解 \((S,\mathfrak{h} )\) 开始，我们构建了一个链图 \(\Phi ^{+}\)，从与\((S,\mathfrak{h})\)相关联的 Heegaard Floer 链复数到由\((S,\mathfrak{h})\)支持的接触形式的内嵌接触同源链复数。链映射 \(\Phi ^{+}\) 与定义在两边的 \(U\)- 映射同调，并还原为定义帽子版本的子复数上的（科林等人，发表于《高等数学研究所》，2024a, 2024b）准同构 \(\Phi\)。代数学的考虑意味着映射 \(\Phi ^{+}\)是一个准同构。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Publications mathématiques de l'IHÉS

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