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$$This page organizes questions and answers about surgery theory. The natural first port of call for quick answers is [http://www.mathoverflow.net/search?q=surgery Mathoverflow]. Below is a list of questions, possibly with answers. The Atlas also has a chapter [[:Category:Questions|Questions]] for questions which attract longer answers. == Questions == === How can you tell if a space is homotopy equivalent to a manifold? === This is in fact a [http://mathoverflow.net/questions/129/how-can-you-tell-if-a-space-is-homotopy-equivalent-to-a-manifold Mathoverflow question]. === Simply connected surgery obstruction groups === <wikitex>; How does one prove that $L_{4j}(e)=\Zz$, $L_{4j+2}(e)=\Zz_2$ and $L_{2k+1}(e) = 0$ ? Read {{cite|Kervaire&Milnor1963}} and/or {{cite|Browder1972}} and/or {{cite|Ranicki2002|Chapter 12}}. </wikitex> === CW structures on topological manifolds === <wikitex>; Topological manifolds of dimension $\leq 3$ have a piecewise lienar (in fact a differentiable) structure, and a fortiori are triangulable. Topological manifolds of dimension $\geq 5$ have a handlebody structure, and hence a $CW$ structure; it is still not known whether there exist non-triangulable examples. It is known that there exist non-triangulable topological 4-manifolds, such as the Freedman $E_8$-manifold. Do topological 4-manifolds have a $CW$ structure? </wikitex> == References == {{#RefList:}} [[Category:Surgery]] [[Category:Questions]]L_{4j}(e)=\Zz$, $L_{4j+2}(e)=\Zz_2$$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 CW structures on topological manifolds

Topological manifolds of dimension $\leq 3$$\leq 3$ have a piecewise lienar (in fact a differentiable) structure, and a fortiori are triangulable. Topological manifolds of dimension $\geq 5$$\geq 5$ have a handlebody structure, and hence a $CW$$CW$ structure; it is still not known whether there exist non-triangulable examples. It is known that there exist non-triangulable topological 4-manifolds, such as the Freedman $E_8$$E_8$-manifold. Do topological 4-manifolds have a $CW$$CW$ 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