Questions about surgery theory

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(Simply connected surgery obstruction groups)
(CW structures on topological manifolds)
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=== CW structures on topological manifolds ===
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=== CW structures on topological 4-manifolds ===
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Topological manifolds of dimension $\leq 3$ have a piecewise lienar (in fact a differentiable)
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Do topological 4-manifolds have a $CW$ structure?
structure, and a fortiori are triangulable. Topological manifolds of dimension $\geq 5$ have a handlebody
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structure, and hence a $CW$ structure; it is still not known whether there exist non-triangulable
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Topological manifolds of dimension $\leq 3$ have a piecewise linear (in fact a differentiable)
examples. It is known that there exist non-triangulable topological 4-manifolds, such as the Freedman
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structure ({{cite|Moise1952}}), and a fortiori are triangulable. Topological manifolds of dimension
$E_8$-manifold. Do topological 4-manifolds have a $CW$ structure?
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$\geq 5$ have a handlebody structure ({{cite|Kirby&Siebenmann1977}}), and hence a $CW$ structure.
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All topological manifolds in dimensions $\geq 5$ can be triangulated if and only if the Kervaire-Milnor-Rohlin
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surjection $\alpha:\theta^H_3\to {\mathbb Z}_2$ splits ({{cite|Galewski&Stern1980}}). There do exist non-triangulable
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topological 4-manifolds, e.g. the Freedman $E_8$-manifold ({{cite|Akbulut&McCarthy1980}}).
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Revision as of 15:00, 15 August 2010

This page organizes questions and answers about surgery theory.

The natural first port of call for quick answers is Mathoverflow.

Below is a list of questions, possibly with answers.

The Atlas also has a chapter Questions for questions which attract longer answers.

Contents

1 Questions

1.1 How can you tell if a space is homotopy equivalent to a manifold?

This is in fact a Mathoverflow question.

1.2 Simply connected surgery obstruction groups

How does one prove that L_{4j}(e)=\Zz, L_{4j+2}(e)=\Zz_2 and L_{2k+1}(e) = 0 ?

Read [Kervaire&Milnor1963] and/or [Browder1972] and/or [Ranicki2002, Chapter 12].

1.3 CW structures on topological 4-manifolds

Do topological 4-manifolds have a CW structure?

Topological manifolds of dimension \leq 3 have a piecewise linear (in fact a differentiable) structure ([Moise1952]), and a fortiori are triangulable. Topological manifolds of dimension \geq 5 have a handlebody structure ([Kirby&Siebenmann1977]), and hence a CW structure. All topological manifolds in dimensions \geq 5 can be triangulated if and only if the Kervaire-Milnor-Rohlin surjection \alpha:\theta^H_3\to {\mathbb Z}_2 splits ([Galewski&Stern1980]). There do exist non-triangulable topological 4-manifolds, e.g. the Freedman E_8-manifold ([Akbulut&McCarthy1980]).

2 References

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