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Results tagged with differential-topology
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user 12156
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”.
49
votes
Accepted
Does there exist any non-contractible manifold with fixed point property?
Take the space $\mathbb{CP}^2$. Its cohomology ring is given by $\mathbb{Z}[a]/a^3$, where $a$ has degree $2$. A map $f:\mathbb{CP}^2\rightarrow \mathbb{CP}^2$ induces a map on the second cohomology g …
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22
votes
Books in advanced differential topology
"Morse theory" by Milnor.
"Lectures on the h-cobordism theorem" by Milnor
"Characteristic classes" by Milnor and Stasheff.
"Topological methods in algebraic geometry" by Hirzebruch
"Fiber bundles" Hus …
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17
votes
Accepted
Oriented vector bundle with odd-dimensional fibers
No.
Let us consider some vector bundles over $S^4$. Since $S^4$ is simply connected all vector bundles are oriented and rank $k$ vector bundles over $S^4$ are classified by $\pi_{3}(SO(k))$ by the cl …
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9
votes
Accepted
The group structure on $[X,S^n]$ induced by the framed bordism
This is an answer to question (2). Let $n=0,1,3,7$ and $i=1,2$ and $d\leq 2n-2$. Let $e=(1,0,\ldots,0)\in S^n$
Let $f_i:X\rightarrow S^n$ be two maps representing framed submanifolds $(M_i,\nu_i)$. Le …
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7
votes
Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic ...
What about something like $f(x,y)=(g(x),y)$ where
$$
g(x)=\begin{cases} x\qquad &x\in(-\infty,1/2]\\
1-x & x\in [1/2,3/2]\\
x-2 & x\in [3/2,\infty)
\end{cases}
$$
$f$ does not induce the identity mapp …
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6
votes
Accepted
Is the action of diffeomorphisms transitive on the set of vector fields with prescribed zero...
Let me summarize the conclusions in the comments. This is rarely true:
There are local obstructions around fixed points: For example the order of vanishing (Andreas Cap), or the Hopf index of an isol …
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6
votes
Accepted
When does a hypersurface have contact-type?
Not all hypersurfaces in $\mathbb{R}^{2n}$ are of contact type.
Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, give …
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6
votes
Accepted
Intersection modulo 2 theory for infinite dimensional manifolds?
One can speak of transersality of intersections in an infinite dimensional context.
If one goes beyond Hilbert manifolds (e.g. Banach, Frechet) one needs to be a bit careful with the definition of tra …
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5
votes
Can we convert any non-vanishing vector field into geodesic field by changing metric?
This is not always possible. For example there exists smooth vector fields on the three sphere without any closed orbit, this is due to Kuperberg. However any geodesic vector field on a closed manifol …
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5
votes
Accepted
restricting the "Whitney" map
No this is false, here is a counter example. Let $f:(\frac{1}{2},1)\times(0,4\pi)\rightarrow \mathbb{R}^2$ be given by $f(r,\theta)=(r \cos(\theta),r\sin(\theta))$. Of course the domain is diffeomorph …
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5
votes
Patching up two trivial fibre bundles induces homology equivalence
I think you can do something like this. Let $Y=D^2$ be the open disc in the plane, $X=S^1$ the circle. Let $X_1$ a point on the circle (the red point below) and $X_2=X\setminus X_1$. Define the map as …
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5
votes
Accepted
Smooth circle actions on Riemannian manifolds and harmonicity of quotient map
I don't think so. I suspect that this action does not exists for generic metric $g$ on manifolds which admit circle actions.
Here is the thing that I have in mind.
Consider the two torus. This clearly …
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4
votes
Accepted
geometric conditions on maps between manifolds inducing monomorphisms on cohomology
Are you looking for things like the following?
Suppose $M$ and $N$ are of the same dimension connected, closed and $\mathbb{Z}/p\mathbb{Z}$ orientable. Suppose that the $(\mathbb{Z}/p\mathbb{Z})$ de …
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4
votes
Accepted
A cobordism theory from Hirsch's "Differential Topology" (reference request)
As noted in the comments this is in Tom Diecks book (section 21.2), and in Wall's differential topology (Section 8.1).
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3
votes
Separating two submanifolds
No, but if the codimension of the two submanifolds is high enough, then it is possible. Generically the intersection $A$ will be a submanifold of dimension $\dim(A)=\dim M+\dim N-\dim X$. Hence if t …
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