Discussion Overview
The discussion revolves around a problem from topology concerning maps from a manifold M of dimension m to a sphere S^p, specifically whether every such map is homotopic to a constant map. The context includes considerations of continuity and differentiability.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant states that if the dimension of M is less than p, every map from M to S^p is homotopic to a constant, referencing a problem from Milnor's topology text.
- Another participant provides an example where the map can indeed be onto, using a space-filling curve to demonstrate a continuous surjection from S^1 to S^2.
- A participant reiterates the original problem and questions whether the map is required to be differentiable.
- Another suggestion is made that perhaps it suffices for the map to be homotopic to a smooth map, referencing Sard's Theorem.
Areas of Agreement / Disagreement
Participants express differing views on the nature of the map (continuous vs. differentiable) and whether it can be onto, indicating that multiple competing views remain in the discussion.
Contextual Notes
There are unresolved assumptions regarding the nature of the maps discussed, particularly concerning continuity and differentiability, as well as the implications of the examples provided.