Extended surgery theory for simply-connected $4k$-manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917594824441856 |
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| author | Nagy, Csaba |
| author_facet | Nagy, Csaba |
| contents | Kreck proved that two $2q$-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q-1)$-smoothings over the same normal $(q-1)$-type $(B,ξ)$. We show that stable diffeomorphism can be replaced by diffeomorphism if the normal smoothings have isomorphic Q-forms (which consists of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism.
As an application, we identify the inertia group of a $(2k-1)$-connected $4k$-manifold with the kernel of a certain bordism map. By the calculations of Senger-Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups.
We also obtain, for a simply-connected $4k$-manifold $M$ with normal $(q-1)$-type $(B,ξ)$ such that $H_q(B)$ is free, an algebraic description of the stable class of $M$, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to $M$. Using this description, we explicitly compute the stable class of manifolds $M$ with rank-$2$ hyperbolic intersection form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extended surgery theory for simply-connected $4k$-manifolds Nagy, Csaba Geometric Topology 57R67 (Primary), 57R19 (Secondary) Kreck proved that two $2q$-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q-1)$-smoothings over the same normal $(q-1)$-type $(B,ξ)$. We show that stable diffeomorphism can be replaced by diffeomorphism if the normal smoothings have isomorphic Q-forms (which consists of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley's Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k-1)$-connected $4k$-manifold with the kernel of a certain bordism map. By the calculations of Senger-Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected $4k$-manifold $M$ with normal $(q-1)$-type $(B,ξ)$ such that $H_q(B)$ is free, an algebraic description of the stable class of $M$, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to $M$. Using this description, we explicitly compute the stable class of manifolds $M$ with rank-$2$ hyperbolic intersection form. |
| title | Extended surgery theory for simply-connected $4k$-manifolds |
| topic | Geometric Topology 57R67 (Primary), 57R19 (Secondary) |
| url | https://arxiv.org/abs/2402.13394 |