Timeline for Topological obstructions to existence of immersion
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 10, 2018 at 21:13 | comment | added | Ian Agol | @BerniWaterman: this can't embed in a compact surface for homological reasons: the homology is infinitely generated, and is represented by loops which intersect in pairs, but the pairs are disjointly embedded (or disjiont genus 1 subsurfaces). Thus each such loop will embed homologically essentially and distinctly into a compact surface, giving infinitely generated homology, a contradiction. One may create analogous examples in any dimension by taking an infinite connect sum of manifolds with non-trivial homology. | |
| Jun 23, 2018 at 6:24 | comment | added | H1ghfiv3 | Interesting. Though intuitively somehow obvious, I don't explicitly understand why it cannot be embedded into a compact manifold. That it can't be "end-compactified" is clear, it has infinite fundamental group. According to Professor Agol's answer, however, it will at least cover, and thus can be immersed into a compact manifold. | |
| Jun 22, 2018 at 15:59 | history | answered | Piotr Hajlasz | CC BY-SA 4.0 |