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. 
[[
include inc:theorem 
title=PoincarΓ© duality
body=The [[$\mathbb{F}_2$]]homology of a compact [[$n$]]dimensional manifold [[$M$]] is symmetric in its grading. That is for all [[$k,n$]], [[$0 \leq k \leq n$]]: [[$$H_k(M; \mathbb{F}_2) \cong H_{nk}(M; \mathbb{F}_2)$$]]
]]
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page revision: 9, last edited: 25 Aug 2023 00:22