Sandbox 3

Theorem Block

Transversality Theorem Theorem. For $M$ and $N$ smooth manifolds and $W \subset N$ a submanifold, the set of smooth transversal maps $f : M \to N$ with $f \pitchfork W$ is residual in $\mathcal{C}^\infty(M,N)$ - the space of all smooth maps from $M$ to $N$. If $W$ is closed, then this set of transverse maps is both open and dense.

box:theorem
theorem

[[
include inc:theorem |
title=Poincaré duality|
body=The [[$\mathbb{F}_2$]]-homology of a compact [[$n$]]-dimensional manifold [[$M$]] is symmetric in its grading. That is for all [[$k,n$]], [[$0 \leq k \leq n$]]: [[$$H_k(M; \mathbb{F}_2) \cong H_{n-k}(M; \mathbb{F}_2)$$]]|
]]

will produce:

Poincaré duality Theorem. The $\mathbb{F}_2$-homology of a compact $n$-dimensional manifold $M$ is symmetric in its grading. That is for all $k,n$, $0 \leq k \leq n$: $$H_k(M; \mathbb{F}_2) \cong H_{n-k}(M; \mathbb{F}_2)$$

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