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.

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 === <wikitex>; 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}}. </wikitex> === Handlebody and CW structures on topological 4-manifolds === <wikitex>; 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): <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> == 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.

2 References

• [Akbulut&McCarthy1990] S. Akbulut and J. D. McCarthy, Casson's invariant for oriented homology $3$$3$-spheres, Princeton University Press, Princeton, NJ, 1990. MR1030042 (90k:57017) Zbl 0695.57011
• [Browder1972] W. Browder, Surgery on simply-connected manifolds, Springer-Verlag, New York, 1972. MR0358813 (50 #11272) Zbl 0543.57003
• [Galewski&Stern1980] D. E. Galewski and R. J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. of Math. (2) 111 (1980), no.1, 1–34. MR558395 (81f:57012) Zbl 0441.57017
• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
• [Kirby&Siebenmann1977] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Princeton University Press, Princeton, N.J., 1977. MR0645390 (58 #31082) Zbl 0361.57004
• [Moise1952] E. E. Moise, Affine structures in $3$$3$-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