# Tuesday, January 11 :::{.remark} References: - - - [PLC's Notes](https://outlookuga-my.sharepoint.com/:f:/g/personal/pbl20394_uga_edu/EjT_H9wRyAhAsZodxgXOgU0BSIYbqWav8X1jZY5v3RxqJA?e=n6dfVJ) Emphasis for the course: applications to low-dimensional topology, lots of examples, and ways to construct contact structures. The first application is critical to 4-manifold theory: ::: ## Application 1 :::{.theorem title="Cerf's Theorem"} Every diffeomorphism $f: S^3\to S^3$ extends to a diffeomorphism $\BB^4\to \BB^4$. ::: :::{.remark} This isn't true in all dimensions! This is essentially what makes Kirby calculus on 4-manifolds possible without needing to track certain attaching data. ::: :::{.remark} There is a standard contact structure on $S^3$: regard $\CC^2 \cong \RR^4$ and suppose $f:S^3\to S^3$. There is an intrinsic property of contact structures called *tightness* which doesn't change under diffeomorphisms and is fundamental to 3-manifold topology. :::{.theorem title="Eliashberg"} There is a unique tight contact structure $\xi_\std$ on $S^3$. ::: So up to isotopy, $f$ fixes $\xi_\std$. ::: :::{.remark} A useful idea: tiling by holomorphic discs. This involves taking $S^1$ and foliating the bounded disc by geodesics -- by the magic of elliptic PDEs, this is unobstructed and can be continued throughout the disc just using convexity near the boundary. In higher dimensions: $\BB^4$ is foliated by a 2-dimensional family of holomorphic discs. ::: ## Application 2 :::{.remark} Another application: monotonic simplification (?) of the unknot. Given a knot $K \injects S^3$, a theorem of Alexander says $K$ can be braided about the $z\dash$axis, which can be described by a word $w\in B_n$, the braid group \[ B_n = \ts{ \sigma_1,\cdots, \sigma_{n-1} \st [\sigma_i, \sigma_j] = 1 \,\abs{i-j}\geq 2,\, \sigma_i \sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i \sigma_{i+1}\, i=1,\cdots, n-2 } .\] This captures positive vs negative braiding on nearby strands, commuting of strands that are far apart, and the Reidemeister 3 move. Write $K = K(\beta)$ for $\beta$ a braid for the braid closure. ::: :::{.remark} Markov's theorem: if $K = K(\beta_1), K(\beta_2)$ where $\beta_1 \in B_n$ and $\beta_2\in B_m$ with $m,n$ not necessarily equal, then there is a sequence of Markov moves $\beta_1$ to $\beta_2$. The moves are: - Stabilization and destabilization: ![](figures/2022-01-11_11-41-29.png) - Conjugation in $B_n$: ![](figures/2022-01-11_11-39-52.png) - Braid isotopy, which preserves braid words in $B_n$. ::: :::{.remark} A theorem of Birman-Menasco: if $K(\beta) = U$ is the unknot for $\beta \in B_n$, then there is a sequence of braids $\ts{\beta_i}_{i\leq k}$ with $\beta_k = 1 \in B_1$ such that - $\beta_i\in B_{n_i}$ - $K(\beta_i) = U$ - $n_1\geq n_2\geq \cdots \geq n_k = 1$ - $\beta_i \to \beta_{i+1}$ is either a Markov move or an exchange move. Here an exchange move is ![](figures/2022-01-11_11-46-31.png) ::: ## Application 3 :::{.remark} Genus bounds. A theorem due to Thurston-Eliashberg: if $\xi$ is either a taut foliation or a tight contact structure on a 3-manifold $Y$ and $\Sigma \neq S^2$ is an embedded orientable surface in $Y$, then there is an Euler class $e(\xi) \in H^2(Y)$. Then \[ \abs{ \inner{ e(\xi)}{ \Sigma } } \leq -g(\Sigma) ,\] which after juggling signs is a lower bound on the genus of any embedded surface. ::: :::{.remark} Taut foliations: the basic example is $F\cross S^1$ for $F$ a surface. The foliation carries a co-orientation, and the tangencies at critical points of an embedded surface will have tangent planes tangent to the foliation, so one can compare the co-orientation to the outward normal of the surface to see if they agree or disagree and obtain a sign at each critical point. Write $c_\pm$ for the number of positive/negative elliptics and $h_\pm$ for the hyperbolics. Then \[ \chi = (e_+ + e_-) - (h_+ + h_-) ,\] by Poincaré-Hopf. On the other hand, $\inner{e(\xi)}{\Sigma} = (e_+ - h_+) - (e_- - h_-)$, so adding this yields \[ \inner{e(\xi)}{\Sigma} +\chi = 2(e_+ - h_+) \leq 0 .\] Isotope the surface to cancel critical points in pairs to get rid of caps/cups so that only saddles remain. ::: ## Contact Geometry :::{.definition title="?"} A contract structure on $Y^{2n+1}$ is a hyperplane field (a codimension 1 subbundle of the tangent bundle) $\xi = \ker \alpha$ such that $\alpha \wedge (d\alpha)\wedgepower{n} > 0$ is a positive volume form. ::: :::{.example title="?"} On $\RR^3$, \[ \alpha = dz - ydx \implies d\alpha = -dy \wedge dx = dx\wedge dy ,\] so \[ \alpha \wedge d\alpha = (dz-ydx)\wedge(dx\wedge dy) = dz\wedge dx\wedge dy = dx\wedge dy\wedge dz .\] ::: :::{.exercise title="?"} On $\RR^5$, set $\alpha = dz-y_1 dx_1 - y_2 dx_2$. Check that \[ \alpha\wedge (d\alpha)^2 = 2(dz \wedge dx_1 \wedge dy_1 \wedge dx_2 \wedge dy_2) .\] :::