# Questions about surgery theory

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Read {{cite|Kervaire&Milnor1963}} and/or {{cite|Browder1972}} and/or {{cite|Ranicki2002|Chapter 12}}. | Read {{cite|Kervaire&Milnor1963}} and/or {{cite|Browder1972}} and/or {{cite|Ranicki2002|Chapter 12}}. | ||

+ | </wikitex> | ||

+ | |||

+ | === CW structures on topological manifolds === | ||

+ | <wikitex>; | ||

+ | Topological manifolds of dimension $\leq 3$ have a piecewise lienar (in fact a differentiable) | ||

+ | structure, and a fortiori are triangulable. Topological manifolds of dimension $\geq 5$ have a handlebody | ||

+ | structure, and hence a $CW$ structure; it is still not known whether there exist non-triangulable | ||

+ | examples. It is known that there exist non-triangulable topological 4-manifolds, such as the Freedman | ||

+ | $E_8$-manifold. Do topological 4-manifolds have a $CW$ structure? | ||

</wikitex> | </wikitex> | ||

## Revision as of 14:43, 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 , and ?

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

### 1.3 CW structures on topological manifolds

Topological manifolds of dimension have a piecewise lienar (in fact a differentiable) structure, and a fortiori are triangulable. Topological manifolds of dimension have a handlebody structure, and hence a structure; it is still not known whether there exist non-triangulable examples. It is known that there exist non-triangulable topological 4-manifolds, such as the Freedman -manifold. Do topological 4-manifolds have a structure?

## 2 References

- [Browder1972] W. Browder,
*Surgery on simply-connected manifolds*, Springer-Verlag, New York, 1972. MR0358813 (50 #11272) Zbl 0543.57003 - [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor,
*Groups of homotopy spheres. I*, Ann. of Math. (2)**77**(1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505 - [Ranicki2002] A. Ranicki,
*Algebraic and geometric surgery*, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001