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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?
Sep
4
comment characteristic classes of homotopy equivalent manifolds
The question as originally stated assumes the two manifolds have different dimensions. If this is what was intended, then the manifolds cannot both be closed, and counterexamples are easy to find. For example, let $M$ be the circle and $N$ the Moebius band (either the compact or the noncompact version). These have different classes $w_1$ since $M$ is an orientable manifold and $N$ is not. Higher-dimensional counterexamples are equally easy to find.
Aug
26
comment Parallelizability of the Milnor's exotic spheres in dimension 7
The stable parallelizability of exotic spheres is Theorem 3.1 of the famous Kervaire-Milnor paper "Groups of homotopy spheres I" in the 1963 Annals. The proof is short but uses several big theorems from the previous decade such as Bott periodicity, the Hirzebruch signature theorem, and Adams' work on the J-homomorphism.
Aug
26
awarded  Nice Answer
Jul
31
comment Manifold approximations to $BO(3)$
For $BO(3)$ one can choose the Grassmannian of 3-planes in ${\mathbb R}^\infty$, and this is in some sense approximated by the finite-dimensional Grassmann manifolds of 3-planes in ${\mathbb R}^n$. What kind of approximation do you have in mind?
Jul
26
answered Essential surfaces in the Exterior of Montesinos knots