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Extendability over the $4$-sphere and invariant spin structures of surface automorphisms
Geometric Topology · Submitted 9 Oct 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that each automorphism of $F_g$ has an invariant spin structure, and obtain a stably extendable result: Each automorphism of $F_g$ is extendable over $S^4$ after a connected sum with the identity map on the torus.
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The abstract, sentence by sentence
It is known that an automorphism of $F_g$, the oriented closed surface of genus $g$, is extendable over the 4-sphere $S^4$ if and only if it has a bounding invariant spin structure \cite{WsWz}. We show that each automorphism of $F_g$ has an invariant spin structure, and obtain a stably extendable result: Each automorphism of $F_g$ is extendable over $S^4$ after a connected sum with the identity map on the torus. Then each automorphism of an oriented once punctured surface is extendable over $S^4$. For each $g\neq 4$, we construct a periodic map on $F_g$ that is not extendable over $S^4$, and we prove that every periodic map on $F_4$ is extendable over $S^4$, which answer a question in \cite{WsWz}. We illustrate for an automorphism $f$ of $F_g$, how to find its invariant spin structures, bounding or not; and once $f$ has a bounding invariant spin structure, how to construct an embedding $F_g\hookrightarrow S^4$ so that $f$ is extendable with respect to this embedding.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It is known that an automorphism of $F_g$, the oriented closed surface of genus $g$, is extendable over the 4-sphere $S^4$ if and only if it has a bounding invariant spin structure \cite{WsWz}. | background 1.00 | 0.03 | 0.06 | asserted 1.00 |
| s1 | We show that each automorphism of $F_g$ has an invariant spin structure, and obtain a stably extendable result: Each automorphism of $F_g$ is extendable over $S^4$ after a connected sum with the identity map on the torus. | result 0.89 | 0.97 | 0.61 | asserted 1.00 |
| s2 | Then each automorphism of an oriented once punctured surface is extendable over $S^4$. | result 0.99 | 0.81 | 0.14 | asserted 1.00 |
| s3 | For each $g\neq 4$, we construct a periodic map on $F_g$ that is not extendable over $S^4$, and we prove that every periodic map on $F_4$ is extendable over $S^4$, which answer a question in \cite{WsWz}. | result 0.92 | 0.95 | 0.19 | asserted 1.00 |
| s4 | We illustrate for an automorphism $f$ of $F_g$, how to find its invariant spin structures, bounding or not; and once $f$ has a bounding invariant spin structure, how to construct an embedding $F_g\hookrightarrow S^4$ so that $f$ is extendable with respect to this embedding. | method 0.72 | 0.93 | 0.07 | asserted 1.00 |
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Original abstract
It is known that an automorphism of $F_g$, the oriented closed surface of genus $g$, is extendable over the 4-sphere $S^4$ if and only if it has a bounding invariant spin structure \cite{WsWz}. We show that each automorphism of $F_g$ has an invariant spin structure, and obtain a stably extendable result: Each automorphism of $F_g$ is extendable over $S^4$ after a connected sum with the identity map on the torus. Then each automorphism of an oriented once punctured surface is extendable over $S^4$. For each $g\neq 4$, we construct a periodic map on $F_g$ that is not extendable over $S^4$, and we prove that every periodic map on $F_4$ is extendable over $S^4$, which answer a question in \cite{WsWz}. We illustrate for an automorphism $f$ of $F_g$, how to find its invariant spin structures, bounding or not; and once $f$ has a bounding invariant spin structure, how to construct an embedding $F_g\hookrightarrow S^4$ so that $f$ is extendable with respect to this embedding.
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What it cites · 6 sentences without a confident label
Uses data or values from 1 paper
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According to [ref] and the data given in [ref] , $f_4,i∈(F_4)$ is extendable over the $3$-sphere if and only if $i=4,9,11,12$.
Extending periodic automorphisms of surfaces to 3-manifolds · Invariant quadratic forms of torsion mapping classes on low genus surfaces · 0.95
Cites as support 1 paper
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(1) follows from [ref] .
Extending periodic maps on surfaces over the 4-sphere · Applications on extending surface automorphisms over $S^4$ · 0.99
Cites as background 1 paper
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On the other hand, the codimension-$1$ extendability is an active topic, see [ref] and the references therein.
Extending periodic automorphisms of surfaces to 3-manifolds · Introduction · 1.00
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