# Questions about surgery theory

(→Simply connected surgery obstruction groups) |
(→CW structures on topological manifolds) |
||

Line 20: | Line 20: | ||

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

− | === CW structures on topological manifolds === | + | === CW structures on topological 4-manifolds === |

<wikitex>; | <wikitex>; | ||

− | Topological manifolds of dimension $\leq 3$ have a piecewise | + | Do topological 4-manifolds have a $CW$ structure? |

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

− | structure, and hence 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 | |

− | $E_8$-manifold. | + | $\geq 5$ 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&McCarthy1980}}). | ||

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

## 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 , and ?

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 structure?

Topological manifolds of dimension have a piecewise linear (in fact a differentiable) structure ([Moise1952]), and a fortiori are triangulable. Topological manifolds of dimension have a handlebody structure ([Kirby&Siebenmann1977]), and hence a structure. All topological manifolds in dimensions can be triangulated if and only if the Kervaire-Milnor-Rohlin surjection splits ([Galewski&Stern1980]). There do exist non-triangulable topological 4-manifolds, e.g. the Freedman -manifold ([Akbulut&McCarthy1980]).

## 2 References

- [Akbulut&McCarthy1980] Template:Akbulut&McCarthy1980
- [Browder1972] W. Browder,
*Surgery on simply-connected manifolds*, Springer-Verlag, New York, 1972. MR0358813 (50 #11272) Zbl 0543.57003 - [Galewski&Stern1980] D. E. Galewski and R. J. Stern,
*Classification of simplicial triangulations of topological manifolds*, Ann. of Math. (2)**111**(1980), no.1, 1–34. MR558395 (81f:57012) Zbl 0441.57017 - [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 - [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann,
*Foundational essays on topological manifolds, smoothings, and triangulations*, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004 - [Moise1952] E. E. Moise,
*Affine structures in -manifolds. V. The triangulation theorem and Hauptvermutung*, Ann. of Math. (2)**56**(1952), 96–114. MR0048805 (14,72d) Zbl 0048.17102 - [Ranicki2002] A. Ranicki,
*Algebraic and geometric surgery*, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001