Questions about surgery theory

(Difference between revisions)
Jump to: navigation, search
(CW structures on topological 4-manifolds)
(CW structures on topological 4-manifolds)
Line 20: Line 20:
</wikitex>
</wikitex>
=== CW structures on topological 4-manifolds ===
+
=== Handlebody and CW structures on topological 4-manifolds ===
<wikitex>;
<wikitex>;
Do topological 4-manifolds have a $CW$ structure?
+
Do topological 4-manifolds have a handlebody structrure? a $CW$ structure?
Topological manifolds of dimension $\leq 3$ have a piecewise linear (in fact a differentiable)
Topological manifolds of dimension $\leq 3$ have a piecewise linear (in fact a differentiable)
Line 30: Line 30:
surjection $\alpha:\theta^H_3\to {\mathbb Z}_2$ splits ({{cite|Galewski&Stern1980}}). There do exist non-triangulable
surjection $\alpha:\theta^H_3\to {\mathbb Z}_2$ splits ({{cite|Galewski&Stern1980}}). There do exist non-triangulable
topological 4-manifolds, e.g. the Freedman $E_8$-manifold ({{cite|Akbulut&McCarthy1990}}).
topological 4-manifolds, e.g. the Freedman $E_8$-manifold ({{cite|Akbulut&McCarthy1990}}).
+
+
Rob Kirby's answer to the handlebody question (28.8.2010): <i>If a TOP handle decomposition, then the first 1-handle attached to a (smooth) 0-handle can be isotoped to have a smooth attaching map, so the result is smooth. Just keep going like this, using the fact that a top embedding of the attaching sphere can be smoothed in dim 3, and the result is a smooth 4-mfd. Some aren't!!!</i> So a nonsmoothable 4-manifold, e.g. the Freedman $E_8$-manifold,
+
cannot have a handlebody structure.
+
</wikitex>
</wikitex>

Revision as of 19:28, 28 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 Handlebody and CW structures on topological 4-manifolds

Do topological 4-manifolds have a handlebody structrure? 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&McCarthy1990]).

Rob Kirby's answer to the handlebody question (28.8.2010): If a TOP handle decomposition, then the first 1-handle attached to a (smooth) 0-handle can be isotoped to have a smooth attaching map, so the result is smooth. Just keep going like this, using the fact that a top embedding of the attaching sphere can be smoothed in dim 3, and the result is a smooth 4-mfd. Some aren't!!! So a nonsmoothable 4-manifold, e.g. the Freedman E_8-manifold, cannot have a handlebody structure.


2 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox