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Results tagged with differential-topology
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user 4298
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”.
2
votes
1
answer
190
views
Uniform smoothness of locally-flat loops
Let M be a smooth Riemannian manifold. Let $f:S^1\to M$ be a locally-flat injective loop. Must there exist such $\varepsilon>0$ that if we connect points $f(0),f(\varepsilon), f(2\varepsilon)... , f([ …
- 5,043
0
votes
degenerating immersion
The answer is no. Two 2-dim smooth immersed in $\mathbb R^3$ objects generically intersect by line, so if intersection is a point then it can be eliminated. But it is clear that near $z^2$ there are n …
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0
votes
2
answers
353
views
Realisability cohomological class as product or as immersed sphere
Let's consider closed simply connected manifold $M^n$ and a $a\in H^k(M)$ and $a*\in H^{n-k}(M)$ is the dual to $a$.
Is it true that dual to $a$ is realisable as a immersed sphere or $ a*=bc $ for s …
- 5,043
6
votes
Accepted
Pairing used in Lefschetz duality
Yes, your formula is right. For the intuitive understanding just compute it for 1- and 2- dimensional half-spaces.
See Bott & Tu, Differential forms in Algebraic topology, $\S 5$, Poincaré duality.
…
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3
votes
2
answers
299
views
Discriminant locus in knot space
Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$.
The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".
Let $f$ be a knot with $n$ double …
- 5,043
2
votes
1
answer
425
views
Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$
Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$?
It is clear that there are singularities which look like intersection of 2 (or 3 or 4) hyper …
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1
vote
Thom polynomial for contact algebraic structures
there is an article of Quo-Shin Chi "The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere" where the dimension of contact curves moduli …
- 5,043
5
votes
3
answers
595
views
Thom polynomial for contact algebraic structures
Let's consider a algebraic contact structure $P$ on $\mathbb CP^3$
and a algebraic curve $C$ degree $d$ and genus $g$. Let's assume
that contact structure has degree $p$ (see
Polynomial contact struct …
- 5,043
10
votes
2
answers
891
views
Sum of two tangent bundles of $S^{2n}$
I was wondering if the sum $TS^{2n}\oplus TS^{2n}$ is a trivial bundle?
The same is true for spheres of odd dimension (one can find a nowhere zero section of the second bundle, add it to the first, th …
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3
votes
1
answer
435
views
First cohomology of the space of long knots in ℝ⁴
Let's consider the space of long knots in $\mathbb R^n$, $n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I th …
- 5,043
1
vote
Accepted
What is the simplest way to show that a section of a vector bundle is transverse to the zero...
It sounds like a multijet transversality theorem (see for example M. Golubitsky, V. Guillemin, Stable mappings and their singularities) in context of algebraic geometry. So, the answer is true --- a p …
- 5,043
7
votes
4
answers
968
views
I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps wi …
- 5,043
5
votes
3
answers
2k
views
Minimal genus, adjunction inequality
Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know …
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