Exotic smoothings via large R^4's in Stein surfaces

Julia Bennett

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Exotic smoothings via large R^4's in Stein surfaces

Julia Bennett

Geometric Topology · Submitted 23 Oct 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.

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What the abstract itself states, as probabilities: claims first 0.92 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.45 · compares with prior results 0.57 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4-manifolds. Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds.	background 0.54	0.75	0.07	asserted 0.80
s1	In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces.	result 0.56	0.96	0.69	asserted 1.00
s2	This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4-manifolds.	method 0.97	0.27	0.05	asserted 0.99
s3	Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.	result 0.97	0.94	0.53	asserted 1.00

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4-manifolds. Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Builds on 1 paper

In particular, we generalize the first statement of Theorem 3.4 from [ref] without using either the genus function or the Taylor invariant, which were both necessary to the proof given in that paper.

Minimal genera of open 4-manifolds · Large $ℝ^4$'s in Stein surfaces · 0.90

Uses the method or tool of 1 paper

These smooth structures will be constructed by end-summing Casson smoothings on $X$ from [ref] with $ℝ^4$'s from Theorem [ref].

Minimal genera of open 4-manifolds · The Taylor invariant, the genus-rank function, and compact equivalence classes · 0.81

Compares with 1 paper

Comparing the previous theorem to Gompf's results from [ref] can sometimes produce two non-diffeomorphic smooth structures on the same underlying topological $4$-manifold, distinguishable by the existence of a smooth, compact, codimension-$0$ submanifold that does not smoothly embed into a Stein surface.

Minimal genera of open 4-manifolds · Large $ℝ^4$'s in Stein surfaces · 1.00

Cites as background 1 paper

More recently, Gompf exploited the rich structure associated to Stein surfaces to study open $4$-manifolds in [ref] .

Minimal genera of open 4-manifolds · Introduction · 0.82

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