Is the square pyramid a manifold with corners?

An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to [0,)n instead of [0,)×Rn1. See also:

The filled cube is naturally a 3-manifold with corners, with every vertex having a neighborhood that is diffeomorphic to a subset of [0,)3. It does not seem possible to me that we can map a neighborhood of the top vertex of the filled square pyramid to [0,)3 since it has four adjacent faces instead of three.

Can we conclude that the filled square pyramid is not a manifold with corners? If yes, what is the point in excluding this seemingly useful structure by such a strict definition?

Answer

If there were a diffeomorphism taking (a neighborhood of) the top vertex of the pyramid to a (neighborhood of) the vertex in [0,)3, taking its derivative it would in particular preserve the tangent cone (the set of tangent vectors vTpM such that v=γ(0) for some curve γ:[0,1]M with γ(0) the vertex). If, for convenience, we set the top point of the pyramid to be 0R3, the tangent cone of the top vertex of the pyramid is the set of scalar multiples of points in the pyramid; the tangent cone of the vertex of the cube is [0,)3. So we’ve reduced this to a linear algebra problem: showing a linear isomorphism cannot take the first cone to the second. But, as you predicted, a linear isomorphism preserves the number of faces of the tangent cone. (I do not want to write down the details, so I leave this to you.)

As for why we don’t want to include the pyramid: I haven’t ever used manifolds with corners much myself, so I can only speculate, but I suspect it’s because much of the interest is in a) developing a theory that works with simplices, so that we can integrate along chains; and b) developing a theory that will allow us to define cobordisms between manifolds with boundaries. Such cobordisms would, near the ‘boundary’, look like M×I, where M is a manifold with boundary; this gives us the local model [0,)2×Rn1. Now we’re going to want cobordisms between these, etc, inductively saying we should want to include all local models [0,)k×Rnk. In doing so we never really needed to have corners like the top of a square pyramid.

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Source : Link , Question Author : Friedrich , Answer Author : Community

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