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**Morse homology and manifolds with boundary.**
*(English)*
Zbl 1123.58006

From the author’s introduction: Let \(M\) be an oriented compact manifold with boundary \(\partial M\). We identify a collar neighborhood of the boundary with \([0, 1) \times M\) and denote by \(r\) the standard coordinate on the first factor. Take a Riemannian metric \(g\) on \(M\backslash\partial M\) such that \(g| _{(0,1)\times \partial M} = \frac{1}{r} dr\otimes dr + rg_{\partial M}\) , where \(g_{\partial M}\) is a Riemannian metric on \(\partial M\); we call such metrics horn end metrics. Let \(f\) be a Morse function on \(M\backslash\partial M\) which satisfies the following conditions: (i) There is a Morse function \(f_{\partial M}\) on \(\partial M\) such that \(f| _{(0,1)\times\partial M} = rf_{\partial M}\). (ii) If \(\gamma\) is a critical point of \(f_{\partial M}\), then \(f_{\partial M} (\gamma)\) is not equal to zero.

We define Morse homology for such Morse functions; the complex is generated by the critical points of \(f\) and the positive ones of \(f_{\partial M}\), and the boundary operator is defined by taking an algebraic count of the isolated descending gradient trajectories between critical points. We prove that the Morse homology is isomorphic to the singular homology of \(M\), and construct the relative Morse homology sequence of \(\partial M\) in \(M\). We remark that our Morse homology is a finite dimensional model of Floer homology for noncompact Lagrangian submanifolds properly embedded into a noncompact symplectic manifold with concave end; the collar neighborhoods correspond to concave ends, and Morse functions and the critical points of their differences correspond to Lagrangian submanifolds and their intersection points, which are the usual correspondences, and moreover boundary Morse functions and the critical points of their differences correspond to Legendrian submanifolds and the Reeb chords between them, and Morse homology of \(\partial M\) corresponds to contact homology.

We define Morse homology for such Morse functions; the complex is generated by the critical points of \(f\) and the positive ones of \(f_{\partial M}\), and the boundary operator is defined by taking an algebraic count of the isolated descending gradient trajectories between critical points. We prove that the Morse homology is isomorphic to the singular homology of \(M\), and construct the relative Morse homology sequence of \(\partial M\) in \(M\). We remark that our Morse homology is a finite dimensional model of Floer homology for noncompact Lagrangian submanifolds properly embedded into a noncompact symplectic manifold with concave end; the collar neighborhoods correspond to concave ends, and Morse functions and the critical points of their differences correspond to Lagrangian submanifolds and their intersection points, which are the usual correspondences, and moreover boundary Morse functions and the critical points of their differences correspond to Legendrian submanifolds and the Reeb chords between them, and Morse homology of \(\partial M\) corresponds to contact homology.

Reviewer: Nils Ackermann (MĂ©xico, D.F.)

### MSC:

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

55N35 | Other homology theories in algebraic topology |

57R70 | Critical points and critical submanifolds in differential topology |

53D40 | Symplectic aspects of Floer homology and cohomology |

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\textit{M. Akaho}, Commun. Contemp. Math. 9, No. 3, 301--334 (2007; Zbl 1123.58006)

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