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Mar
28 |
comment |
Homotopy groups of Lie groups
@Ben Wieland: The vector space needs to be of countable dimension in order to have $CP^\infty$ as its projectivization. However, if the rational functions had a countable basis, this would imply that only countably many complex numbers occurred as poles of rational functions, a contradiction. This is almost counterintuitive: rational functions depend upon only countably many complex parameters (their coefficients) yet as a vector space their dimension is uncountable. |
Mar
19 |
awarded | Nice Answer |
Feb
23 |
awarded | Enlightened |
Feb
23 |
awarded | Nice Answer |
Feb
9 |
awarded | Informed |
Dec
1 |
comment |
hyperbolic structure on Figure–8 knot complement
I seem to recall that Bill Thurston told me that these ideal triangulations of punctured-torus bundles are due to Troels Jorgensen. The earliest appearance of them in the literature that I am aware of is in a paper by Bill Floyd and myself, "Incompressible surfaces in punctured-torus bundles", Top. and its Appl. 13 (1982), 263-282. Gueritaud's paper seems to have a nice exposition, with nicer pictures than the hand-drawn ones in my paper with Floyd. |
Nov
13 |
awarded | Yearling |
Nov
9 |
comment |
when is “fibering” preserved under homotopy equivalence
@ThiKu. Of the 28 exotic 7-spheres, only 16 are fiber bundles over $S^4$ with fiber $S^3$. I'm not sure who originally noted this, but it can be deduced from a paper of Itiro Tamura, "Remarks on differentiable structures on spheres" in J. Math. Soc. Japan. 13 (1961), 383-386. |
Nov
5 |
answered | Equivalence relations of topological spaces not comparable with homotopy |
Nov
1 |
answered | positions of regular cubes in Euclidean space with all its vertices without distinction |
Oct
26 |
answered | Homotopy classes of homeomorphisms of a multiple pointed space |
Oct
12 |
comment |
positions of regular cubes in Euclidean space with all its vertices without distinction
This is a closed orientable 3-manifold so its homology groups are determined by its fundamental group, which is the binary octahedral group. There is a very nice book by Montesinos that covers all examples of this type in three dimensions, called "Classical Tessellations and Three-Manifolds". |
Oct
9 |
awarded | Nice Answer |
Oct
8 |
answered | Topological structure of SO(n) as a product |
Oct
4 |
awarded | Good Answer |
Oct
3 |
comment |
Does every connected component of a covering space over a connected base intersect all the fibers of the covering space?
The question should be rephrased to eliminate the ambiguity of what the word "its" refers to. (Does it refer to "connected component" or to "covering space"?) |
Sep
25 |
answered | Cup product of cohomology in a Serre spectral sequence |
Sep
23 |
comment |
Are there non-contractible spaces $A$ and $B$ such that $A \wedge B$ is contractible?
In the same spirit, just take the smash product of two CW complexes that are Moore spaces for finite cyclic groups of relatively prime orders. |
Sep
21 |
revised |
What are the possible automorphism groups of Riemann surfaces of low genus?
added 724 characters in body |
Sep
21 |
answered | What are the possible automorphism groups of Riemann surfaces of low genus? |