Discussion Overview
The discussion centers around the orientability of non-smooth C^1 knots, specifically those classified as S^1-knots, which are homeomorphisms of S^1 into R^3 or S^3. Participants explore the implications of smoothness on orientability and the definitions of orientability in various contexts, including the use of homology.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions whether C^1 knots can be guaranteed to be orientable without assuming smooth embeddings, suggesting that if S^1 is embedded, it is a submanifold with a trivial normal bundle, which implies orientability.
- Another participant argues that C^1 suffices for providing a continuous tangent vector field along the curve, which may support orientability.
- A participant seeks a more geometrically intuitive definition of orientable bundles beyond the reduction of the structure group to O(n), discussing various definitions involving non-zero sections and cohomology classes.
- Some participants note that a closed curve without self-intersection in a manifold is likely to be embedded, although C^1 does not guarantee the absence of self-intersection.
- There is a discussion about whether a homeomorphism induces an isomorphism in homology, with some participants asserting that it does, while others caution that embedding one manifold in another may lead to zero homology classes.
- One participant reflects on the implications of homology classes being zero in Euclidean space for knots, raising questions about the top homology when embedding S^1 in Euclidean n-space.
Areas of Agreement / Disagreement
Participants express differing views on the implications of C^1 smoothness for orientability, the definitions of orientability, and the effects of homeomorphisms on homology. The discussion remains unresolved with multiple competing perspectives presented.
Contextual Notes
Participants highlight limitations in definitions and assumptions regarding smoothness and embeddings, as well as the potential for different homological outcomes based on the context of the embeddings.