Twisted Morse complexes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866917892667211776 |
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| author | Banyaga, Augustin Hurtubise, David Spaeth, Peter |
| author_facet | Banyaga, Augustin Hurtubise, David Spaeth, Peter |
| contents | In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with local coefficients for regular CW-complexes, and singular homology with local coefficients. By proving Morse theoretic versions of the Poincare Lemma and of the de Rham Theorem, we show that twisted Morse cohomology with coefficients in a local system determined by a closed 1-form is isomorphic to the Lichnerowicz cohomology obtained by deforming the de Rham differential by the 1-form. We demonstrate the effectiveness of twisted Morse complexes by using them to compute Lichnerowicz cohomology, to compute obstructions to spaces being associative H-spaces, and to compute Novikov numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_07818 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Twisted Morse complexes Banyaga, Augustin Hurtubise, David Spaeth, Peter Algebraic Topology Dynamical Systems Geometric Topology Symplectic Geometry Primary: 58E05 Secondary: 37D15 55N25 55P45 In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with local coefficients for regular CW-complexes, and singular homology with local coefficients. By proving Morse theoretic versions of the Poincare Lemma and of the de Rham Theorem, we show that twisted Morse cohomology with coefficients in a local system determined by a closed 1-form is isomorphic to the Lichnerowicz cohomology obtained by deforming the de Rham differential by the 1-form. We demonstrate the effectiveness of twisted Morse complexes by using them to compute Lichnerowicz cohomology, to compute obstructions to spaces being associative H-spaces, and to compute Novikov numbers. |
| title | Twisted Morse complexes |
| topic | Algebraic Topology Dynamical Systems Geometric Topology Symplectic Geometry Primary: 58E05 Secondary: 37D15 55N25 55P45 |
| url | https://arxiv.org/abs/1911.07818 |