Error With Counterexample
Last week, I proposed an potential graph that would improve a bound on Grunbaum’s conjecture. However, I now see that I missed a defect on the inside of the embedding. Unfortunately, it doesn’t look like this will be easy to fix, so I’m going to dedicate the coming week reading more papers and looking at more techniques.
Currently, I am reading more on the theory behind superposition, which has led me to learning more about nowhere zero flows in a paper by Kochol. I am currently reading through the paper and trying to get a better understanding of the concept.
One potential solution that I have been thinking about would be to define a new graph operation that preserves snarkiness. This operation should take two non-adjacent vertices x and y from the first graph and two non-adjacent vertices u and v, such that x connects to u and y connects to v. Operations I am familiar with like the dot product or star product don’t have this property unfortunately. If I could create such an operation, then I should be able to construct a counterexample to Grunbaum’s conjecture of genus 4.
Progress With Finite Spaces
It turns out that the minimum number of points to construct a space with homology of the sphere is known, and we need 2n+2 points to make a space with homology Sn (the n-dimensional sphere). Another direction I am considering is, given a group G, construct a finite space with nth homology group G. I read a similar result using Moore spaces (from Hatcher), which are constructed by attaching n-cells (thought of as n-dimensional balls) rather than finite spaces. The method of approach would be to use a result by McCord that shows an equivalence between finite spaces and simplicial complexes.
- Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.
- J.P. May. (N.D.). Finite Spaces And Larger Contexts. Department Of Mathematics | The University Of Chicago. Https://Math.Uchicago.Edu/~May/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.Pdf
- Kochol, M. (2002). Superposition And Constructions Of Graphs Without Nowhere-Zero K-Flows. European Journal Of Combinatorics, 23(3), 281-306. Https://Doi.Org/10.1006/Eujc.2001.0563
- McCord, Michael C. (1966). “Singular Homology Groups And Homotopy Groups Of Finite Topological Spaces” (PDF). Duke Math. J. 33 (3): 465–474. Doi:10.1215/S0012-7094-66-03352-7