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Results tagged with differential-topology
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user 12310
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
59
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3
answers
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Operations via Morse Theory
I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology operat …
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39
votes
4
answers
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How to tackle the smooth Poincaré conjecture
The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. Freed …
- 17.5k
26
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Vector fields on $(4n+1)$-spheres
If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, what is the maximum numbe …
- 17.5k
16
votes
1
answer
687
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Study topology from existence of multiple smooth structures?
There are classical existence results of a smooth structure on a topological manifold, and many results on the existence of multiple (i.e. exotic) smooth structures. Some utilize Freedman's theorem in …
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10
votes
2
answers
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Floer homology and Invariants for Einstein Field Equations?
Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, leading to the Donaldson inva …
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9
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When does a hypersurface have contact-type?
In a symplectic manifold $(X^{2n},\omega)$, a hypersurface $Y\subset X$ has contact-type if there is a contact form $\lambda$ such that $d\lambda=\omega|_Y$. Recall that a contact form is a 1-form wit …
- 17.5k
8
votes
1
answer
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Milnor immersion of circle, disks, and a ball
Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [1] for example, picture included). I think I can glue these two disks together to d …
- 17.5k
5
votes
1
answer
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Conjecture on homotopy groups of moduli space of self-dual connections
In the paper Stability in Yang-Mills Theories (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured:
"The inclusion $\mathcal{M}_n\hookrightar …
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5
votes
1
answer
270
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Irreducibility of 3-manifolds with (non)empty boundary
All manifolds considered here are compact and orientable. A 3-manifold (with possible boundary) is irreducible if any smooth sphere bounds a ball. Note that a closed irreducible 3-manifold is prime, a …
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4
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1
answer
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Spin-c Structures viewed w.r.t. Cell Decomposition
In my quest to understand spin representations, I am looking at the equivalent views of spin structures (on some oriented Riemannian $n$-manifold). Given such a manifold $M$, its tangent bundle $TM$ …
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4
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1
answer
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Concerning strata in $C^\infty(M)$
The Morse functions are dense in $C^\infty(M)$, and you can ask if a 1-parameter family of smooth functions between two given Morse functions will be a homotopy through Morse functions. Well, Cerf The …
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