---
date: 2022-02-23 18:45
modification date: Monday 4th April 2022 09:37:23
title: "surgery"
aliases: [surgery theory]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #geomtop 
- Refs:
	- Lück's Basic introduction to surgery theory
	- Recommended by Akram
		- Using surgery theory to study [homotopy sphere](homotopy%20sphere) : [attachments/1970128.pdf](attachments/1970128.pdf) 
		#resources/papers #resources/recommendations
		- Killing homotopy smoothly: [paper from Milnor](https://www.maths.ed.ac.uk/~v1ranick/papers/milnorsurg.pdf) 
		#resources/papers #resources/recommendations 
- Links:
	- [L theory](L%20theory)
	- [h-cobordism](h-cobordism.md)
	- [Surgery Classification.svgz](Surgery%20Classification.svgz)

---

# Surgery

# Motivation: CW Cell Attachment

Given $X$, we can form $\tilde X = X^n \disjoint_\phi e^n$ where $e^n \cong \DD^n$ is an $n\dash$cell and $\phi: S^{n-1} \to X$ is the characteristic/attaching map.

> Remark: Why $S^{n-1}$? This just comes from the fact that $\bd e^n = \bd \DD^n = S^{n-1}$.

![](2020-02-05-00-22-18.png)

**Problem**:
This doesn't "see" the [smooth structure](smooth%20structure.md), and CW complexes can have singular points, e.g. $S^2 = e^0 \disjoint e^2$.

**Solution**:
Use [tubular neighborhood](tubular%20neighborhood), for each sphere, thicken with a disc of its codimension.

# Definitions

**Definition (Surgery):**
Given a manifold $M^n$ where $n=p+q$, then $p\dash$surgery on $M$, denoted $\mathcal{S}(M)$, result of cutting out $S^p \cross D^q$ and gluing back in $D^{p+1} \cross S^{q-1}$.

Let $\Gamma_{p, q} = S^p \cross D^q$, call this our "surgery cell".
As in the CW case, we want to attach this cell via an embedding of its boundary into $M$.

We can compute
$$
\bd(S^p\cross D^q) = S^p \cross S^{q-1} = \bd(\mathbf{D^{p+1} \cross S^{q-1}})
$$ 

then the above says
$$
\bd \Gamma_{p, q} = S^p \cross S^q = \bd \Gamma_{p+1, q-1}
$$

So fix any [embedding](embedding.md) 
$$\phi: \Gamma_{p, q} \to M$$

Note that this restricts to some map (abusing notation) 
$$\phi: \bd \Gamma_{p, q} \to M$$

So by the above observation, we can trade this in for a map
\[
\phi: \bd\Gamma_{p+1, q-1} \to M
.\]

And so we can use this as an attaching map:
\[
\mathcal{S}_p(M) \definedas M\setminus \phi(\Gamma_{p, q})^\circ \disjoint_\phi \Gamma_{p+1, q-1} 
.\]


**Definition ([Handle Attachment](Handle%20Attachment))**
Given a manifold $(M^n, \bd M^n)$ with boundary, attaching a $p\dash$[handle](handle) to $M$, denoted $H_p(M)$, is given by $p\dash$surgery on $\bd M$, i.e. 
\[
H_p(M)^\circ    &= M \\
\bd H_p(M)      &= \mathcal{S}_k(\bd M)
.\]

Remark: we need conditions on the embedding of the [normal bundle](normal%20bundle.md) for this to work.

# Examples

**Examples of Handles :**
$S^1 \cross D^2 \cong \bar T$, a solid torus.

A useful table:

![](2020-02-05-00-59-19.png)

![](2020-02-05-12-25-15.png)



# Results

- Every compact manifold is surgery on a [link](link) and admits a [contact structure](contact%20structure).


![](attachments/Pasted%20image%2020220404093723.png)

# Kervaire invariant

Relation to [Kervaire invariant one](Unsorted/Kervaire%20invariant%201.md):

![](attachments/Pasted%20image%2020220404093950.png)