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at.algebraic-topology
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- 31 votes cast
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 |
Jul
14 |
awarded | Good Answer |