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Topology is a branch of mathematics that studies the properties of spaces and their relationships. In topology, the emphasis is on the geometric properties that are preserved under continuous transformations, such as stretching, bending, and twisting.
A topological space is a set of points, along with a collection of open sets that satisfy certain axioms. These open sets are subsets of the space that are considered “open” because they contain the points around them. The open sets form the basis for defining concepts like continuity, convergence, and connectedness.
Topology is used in many areas of mathematics and science, including geometry, analysis, algebraic topology, and computer science. It has applications in physics, engineering, and other fields where understanding the shape and structure of spaces is important.
Let $X, Y \in C^{\infty}(T)$ and $\pi \in C^{\infty}\left(\wedge^2 T\right)$, so that, in a coordinate patch with coordinates $x_i$, we have $X=X^i \frac{\partial}{\partial x^i}, Y=Y^i \frac{\partial}{\partial x^i}$ and $\pi=\pi^{i j} \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial x^j}$. Compute $[X, Y],[\pi, X]$, and $[\pi, \pi]$ in coordinates.
We begin by computing the Lie bracket $[X,Y]$ of the vector fields $X$ and $Y$: \begin{align*} [X,Y] &= X(Y) – Y(X) \ &= X^i\frac{\partial Y^j}{\partial x^i}\frac{\partial}{\partial x^j} – Y^i\frac{\partial X^j}{\partial x^i}\frac{\partial}{\partial x^j} \ &= \left(X^i\frac{\partial Y^j}{\partial x^i} – Y^i\frac{\partial X^j}{\partial x^i}\right)\frac{\partial}{\partial x^j}. \end{align*} So, in coordinates, we have
$[X, Y]=\left(X^i \frac{\partial Y^j}{\partial x^i}-Y^i \frac{\partial X^j}{\partial x^i}\right) \frac{\partial}{\partial x^j}$.
Next, we compute the Lie bracket $[\pi, X]$ of the 2-form $\pi$ and the vector field $X$. We have \begin{align*} [\pi,X] &= \pi(X,\cdot) \ &= \pi^{ij}\frac{\partial X^k}{\partial x^i}\frac{\partial}{\partial x^j}\wedge\frac{\partial}{\partial x^k} \ &= \pi^{ij}\frac{\partial X^k}{\partial x^i}\frac{\partial}{\partial x^j k}. \end{align*} So, in coordinates, we have
$[\pi, X]=\pi^{i j} \frac{\partial X^k}{\partial x^i} \frac{\partial}{\partial x^j k}$
Finally, we compute the Lie bracket $[\pi,\pi]$ of the 2-form $\pi$ with itself. We have \begin{align*} [\pi,\pi] &= 2\pi\wedge\pi \ &= 2\pi^{ij}\pi^{kl}\frac{\partial}{\partial x^i}\wedge\frac{\partial}{\partial x^j}\wedge\frac{\partial}{\partial x^k}\wedge\frac{\partial}{\partial x^l} \ &= 2\left(\pi^{ij}\pi^{kl}-\pi^{il}\pi^{kj}\right)\frac{\partial}{\partial x^i}\wedge\frac{\partial}{\partial x^j}\wedge\frac{\partial}{\partial x^k}\wedge\frac{\partial}{\partial x^l}. \end{align*} So, in coordinates, we have
$[\pi, \pi]=2\left(\pi^{i j} \pi^{k l}-\pi^{i l} \pi^{k j}\right) \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial x^j} \wedge \frac{\partial}{\partial x^k} \wedge \frac{\partial}{\partial x^l}$.
Show that $S^4$ has no symplectic structure. Show that $S^2 \times S^4$ has no symplectic structure.
To show that $S^4$ has no symplectic structure, we can use the fact that every closed 2-form on a compact, connected 4-manifold $M$ satisfies $[\omega]^2 = 2\chi(M)$, where $[\omega]$ denotes the cohomology class of the 2-form $\omega$, and $\chi(M)$ is the Euler characteristic of $M$. In particular, if $M$ admits a symplectic structure, then $[\omega]$ is a nonzero multiple of the generator of $H^2(M;\mathbb{Z})$, and so $[\omega]^2$ is positive. However, we have $\chi(S^4) = 2$, so any closed 2-form on $S^4$ must satisfy $[\omega]^2 = 4$. This is a contradiction, so $S^4$ cannot admit a symplectic structure.
To show that $S^2 \times S^4$ has no symplectic structure, we can use the fact that the product of two symplectic manifolds is symplectic. If $S^2 \times S^4$ had a symplectic structure, then in particular both factors $S^2$ and $S^4$ would admit symplectic structures. However, we just showed that $S^4$ does not admit a symplectic structure, so $S^2 \times S^4$ cannot admit a symplectic structure either.
Write the Poisson bracket $\{f, g\}$ in coordinates for $\pi=\pi^{i j} \frac{\partial}{\partial x^i} \wedge$ $\frac{\partial}{\partial x^3}$
The Poisson bracket of two functions $f$ and $g$ is defined as:
$${f, g} = \pi(df,dg) = \frac{\partial f}{\partial x^i}\pi^{ij}\frac{\partial g}{\partial x^j}-\frac{\partial g}{\partial x^i}\pi^{ij}\frac{\partial f}{\partial x^j}$$
where $\pi^{ij}$ are the components of the Poisson bivector $\pi$.
In coordinates, we have $\pi=\pi^{ij}\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial x^j}=\pi^{12}\frac{\partial}{\partial x^1}\wedge \frac{\partial}{\partial x^2}+\pi^{13}\frac{\partial}{\partial x^1}\wedge \frac{\partial}{\partial x^3}+\pi^{23}\frac{\partial}{\partial x^2}\wedge \frac{\partial}{\partial x^3}$, where $\pi^{ij}$ are functions of the coordinates $x^1,x^2,x^3$.
Using this expression for $\pi$, we can compute the Poisson bracket ${f, g}$ for any functions $f$ and $g$.
