Sergei Yakovenko's blog: on Math and Teaching

Wednesday, January 11, 2017

Lecture 9, Jan 2, 2017

Sundries

I briefly discussed the (simplicial) homology construction in application to smooth manifolds and described several pairings: de Rham pairing (integration) between homology and cohomology, intersection form between H_k(M^n,\mathbb Z) and H_{n-k}(M^n,\mathbb Z), the pairing H^k_\text{dR}(M^n,\mathbb R)\times H^{n-k}_\text{dR}(M^n,\mathbb R)\to\mathbb R, \quad (\alpha,\beta)\longmapsto \displaystyle \int_M \alpha\land\beta and the Poincare duality.

Then I mentioned without proofs several results stressing the role of smoothness, in particular, how different smooth structures can live on homeomorphic manifolds. The tale of planar curve eversion and sphere eversion was narrated. For the video of the sphere eversion go here.

In the second part of the lecture I discussed natural additional structures that can live on smooth manifolds, among them

  • Complex structure, almost complex structure,
  • Symplectic structure,
  • Parallel transport,
  • Riemannian metric,
  • Group structure.

Then we prepared the ground for the next lecture, discussing how examples of these structures naturally appear (e.g., on submanifolds of the Euclidean space, on quotient spaces, …)

There will be no notes for this lecture, because of its mostly belletristic style.

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