By Dieudonne J.
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Take a torus and start to pinch a certain circle on it, until this circle is just a point – this is the critical frame of the ﬁlm – and continue: the two sides of the pinching circle break apart, and the whole surface becomes the skin of a sausage, which is then gradually contracted until it is a sphere. 4 there is a frame with a singular manifold – is that a problem? No, this is normal, and in fact necessary in order to make a transition between two manifolds which are not diffeomorphic (cf. 8).
This result goes back to the 1920s (Rado, Ker´ekjart´o, . . 14 The composition of two cobordism classes. So far we have shown that given speciﬁc cobordisms M0 : 0 1 and M1 : 1 2 then there is a well deﬁned diffeomorphism class M0 M1 : 0 2. We must now check that the result does not depend on the actual cobordisms chosen, but only on their class, so that it makes sense to compose cobordism classes. But in a indirect way we have already shown that. Suppose we have ∼ M and ψ : M → ∼ M , diffeomorphisms (rel the boundary) ψ0 : M0 → 1 1 0 1 ✲ 0 ✲ M0 ✛ ✻ ψ ✲ 1 0 M0 ✲ ✛ M1 ✛ ✻ ψ1 M1 2 ✛ then there is a gluing M0 M1 and a gluing M0 M1 , and also the two diffeomorphisms ψ0 and ψ1 glue in the category of continuous maps, so we get ∼ M M which is a diffeomorphism on each a homeomorphism ψ : M0 M1 → 0 1 piece ✲ M0 M1 ✻ ✛ ψ homeo.
But this is all guaranteed by the universal property. First, the universal U1 (with Rn in the place of X) implies there is a unique property of U = U0 continuous map U → Rn making the diagram commute. Second, the universal 38 Cobordisms and TQFTs property of Rn provides a continuous map in the other direction, and clearly they are inverses to each other, so f is indeed a homeomorphism. So we have constructed a coordinate chart with domain U . Now there were choices involved: for each choice of f0 and f1 , the construction gives a chart f on U .
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