# Questions about surgery theory

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 Handlebody and CW structures on topological 4-manifolds

Do topological 4-manifolds have a handlebody structrure? 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&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 -manifold,
cannot have a handlebody structure.

## 2 References

- [Akbulut&McCarthy1990] S. Akbulut and J. D. McCarthy,
*Casson's invariant for oriented homology -spheres*, Princeton University Press, Princeton, NJ, 1990. MR1030042 (90k:57017) Zbl 0695.57011 - [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