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• Tags
  • #geomtop
• Refs:
  • Lück’s Basic introduction to surgery theory
  • Recommended by Akram
    • Using surgery theory to study homotopy sphere : attachments/1970128.pdf #resources/papers #resources/recommendations
    • Killing homotopy smoothly: paper from Milnor #resources/papers #resources/recommendations
• Links:
  • L theory
  • h-cobordism
  • Surgery Classification.svgz

Surgery

Motivation: CW Cell Attachment

Given \(X\), we can form \(\tilde X = X^n {\textstyle\coprod}_\phi e^n\) where \(e^n \cong {\mathbb{D}}^n\) is an \(n{\hbox{-}}\)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 \({{\partial}}e^n = {{\partial}}{\mathbb{D}}^n = S^{n-1}\).

Problem: This doesn’t “see” the smooth structure, and CW complexes can have singular points, e.g. \(S^2 = e^0 {\textstyle\coprod}e^2\).

Solution: Use tubular neighborhood, for each sphere, thicken with a disc of its codimension.

Definitions

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

Let \(\Gamma_{p, q} = S^p \times D^q\), 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 \begin{align*} {{\partial}}(S^p\times D^q) = S^p \times S^{q-1} = {{\partial}}(\mathbf{D^{p+1} \times S^{q-1}}) \end{align*}

then the above says \begin{align*} {{\partial}}\Gamma_{p, q} = S^p \times S^q = {{\partial}}\Gamma_{p+1, q-1} \end{align*}

So fix any embedding \begin{align*}\phi: \Gamma_{p, q} \to M\end{align*}

Note that this restricts to some map (abusing notation) \begin{align*}\phi: {{\partial}}\Gamma_{p, q} \to M\end{align*}

So by the above observation, we can trade this in for a map \begin{align*} \phi: {{\partial}}\Gamma_{p+1, q-1} \to M .\end{align*}

And so we can use this as an attaching map: \begin{align*} \mathcal{S}_p(M) \coloneqq M\setminus \phi(\Gamma_{p, q})^\circ {\textstyle\coprod}_\phi \Gamma_{p+1, q-1} .\end{align*}

Definition (Handle Attachment) Given a manifold \((M^n, {{\partial}}M^n)\) with boundary, attaching a \(p{\hbox{-}}\)handle to \(M\), denoted \(H_p(M)\), is given by \(p{\hbox{-}}\)surgery on \({{\partial}}M\), i.e.  \begin{align*} H_p(M)^\circ &= M \\ {{\partial}}H_p(M) &= \mathcal{S}_k({{\partial}}M) .\end{align*}

Remark: we need conditions on the embedding of the normal bundle for this to work.

Examples

Examples of Handles : \(S^1 \times D^2 \cong \mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu\), a solid torus.

A useful table:

Results

• Every compact manifold is surgery on a link and admits a contact structure.

Kervaire invariant

Relation to Kervaire invariant one: