# Questions about surgery theory

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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.

## 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$$This page organizes questions and answers about surgery theory. The natural first port of call for quick answers is [http://www.mathoverflow.net/search?q=surgery Mathoverflow]. Below is a list of questions, possibly with answers. The Atlas also has a chapter [[:Category:Questions|Questions]] for questions which attract longer answers. == Questions == === How can you tell if a space is homotopy equivalent to a manifold? === This is in fact a [http://mathoverflow.net/questions/129/how-can-you-tell-if-a-space-is-homotopy-equivalent-to-a-manifold Mathoverflow question]. === 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 {{cite|Kervaire&Milnor1963}} and/or {{cite|Browder1972}} and/or {{cite|Ranicki2002|Chapter 12}}. === 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 ({{cite|Moise1952}}), and a fortiori are triangulable. Topological manifolds of dimension \geq 5 have a handlebody structure ({{cite|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 ({{cite|Galewski&Stern1980}}). There do exist non-triangulable topological 4-manifolds, e.g. the Freedman E_8-manifold ({{cite|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. == References == {{#RefList:}} [[Category:Surgery]] [[Category:Questions]]L_{4j}(e)=\Zz$, $L_{4j+2}(e)=\Zz_2$$L_{4j+2}(e)=\Zz_2$ and $L_{2k+1}(e) = 0$$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$$CW$ structure?

Topological manifolds of dimension $\leq 3$$\leq 3$ have a piecewise linear (in fact a differentiable) structure ([Moise1952]), and a fortiori are triangulable. Topological manifolds of dimension $\geq 5$$\geq 5$ have a handlebody structure ([Kirby&Siebenmann1977]), and hence a $CW$$CW$ structure. All topological manifolds in dimensions $\geq 5$$\geq 5$ can be triangulated if and only if the Kervaire-Milnor-Rohlin surjection $\alpha:\theta^H_3\to {\mathbb Z}_2$$\alpha:\theta^H_3\to {\mathbb Z}_2$ splits ([Galewski&Stern1980]). There do exist non-triangulable topological 4-manifolds, e.g. the Freedman $E_8$$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$$E_8$-manifold, cannot have a handlebody structure.