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In dimensions ≥5, topology and algebra match up simply. In dimensions 3 and 4, manifolds are more interesting. We used to wonder if there was some key idea from high-dimensional manifolds that would extend at least down to dimension 4, making 4-manifolds easy to understand. There is such a principle: Imbedding 2-handles in smooth 4-dimensional manifolds-with-boundary subject to a homotopy criterion. This high-dimensional idea does work in 4 dimensions and give us insight into 4-manifolds.
the topology of 4-manifolds by michael hartley freedman
(heavily reworded by me)
Linear maps vs diffeomorphisms
diffeomorphic things
by Guillemin & Pollack, Differential Topology
diffeomorphisms said: So pretty!!! <3
Thankyou!! <3
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Any smooth compact manifold is diffeomorphic to the configuration space of some planar linkage.
pictures by Robert Ghrist and Kevin Walker
By the way: this is how windscreen wipers work. (last picture)
