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Results tagged with differential-topology
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user 9928
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”.
107
votes
Parallelizability of the Milnor's exotic spheres in dimension 7
A much more general result is true.
Theorem: Let $\Sigma$ be a homotopy sphere and $f: S^n \to \Sigma $ be a homotopy equivalence. Then $f^{\ast} T \Sigma \cong T S^n$.
It says that exotic spheres c …
- 20.9k
66
votes
Accepted
Is there a sheaf theoretical characterization of a differentiable manifold?
Definition: ''A smooth manifold is a locally ringed space $(M;C^{\infty})$ which satisfies the conditions:
Each $x \in M$ admits a neighborhood $U$, such that $(U,C^{\infty})$ is isomorphic to $(\ma …
- 20.9k
22
votes
Euler characteristic of orbifolds
As far as I understand your question, you want to see a derivation of the formula for $\chi(M/G)$. Here it is:
The difficult part of the argument os to show that there is an isomorphism $H^* (M/G; …
- 20.9k
14
votes
Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
Kosinski, ''Differential manifolds'', Chapter VII, section 2. He gives a detailed proof in the case of just one critical point.
- 20.9k
9
votes
Nice things that can be proved easily with characteristic classes
I like this example. The Spheres $S^{2n}$ cannot be complex manifolds unless $n=0,1,3$.
One proves that $TS^{2n}$ does not have the structure of a complex vector bundle in these cases. If $TS^{2n}$ we …
- 20.9k
8
votes
Betti number and harmonic forms
If I understood your question correctly, it is: how can I compute the dimension of the space of harmonic forms?
There is one class of Riemann manifolds where it is possible to write down the harmonic …
- 20.9k
7
votes
Submersions of closed manifolds
Phillips theorem is plainly wrong in the compact case, and for fairly non subtle reasons.
Take a closed oriented $3$-manifold $M$. There are plenty of bundle epimorphisms $TM \to \mathbb{R}$ because …
- 20.9k
6
votes
Accepted
Generalizations of the handle trading techniques
You might find the paper by C.T.C Wall: Geometrical connectivity I, J. London Math. Soc. 3 (1971), p. 597-604, interesting.
What Wall proves, entirely by handle trading, is that if $W:M_0 \to M_1$ is …
- 20.9k
6
votes
Accepted
Euler class of S^1-orbibundle
Here is a topological construction of such a class, in singular cohomology with rational coefficients.
Let $M$ be an $S^1$-space. Then there is the Borel construction $M // S^1 := ES^1 \times_{S^1} M …
- 20.9k
5
votes
Computing the Euler characteristic of the complex projective plane using differential topology
Take a $3 \times 3$ complex diagonal matrix $A$ with distinct nonzero diagonal entries. The 1-parameter subgroup $exp(At)$ acts on $CP^2$; the fixed points are the lines in $C^3$ containing eigenvecto …
- 20.9k
5
votes
Topological degree theory
If $D$ is a smooth compact manifold with boundary, then any $f: \partial D \to \mathbb{R}^n \setminus 0$ of degree $0$ can be extended to a map $D \to \mathbb{R}^n \setminus 0$. Proof: A theorem of Ho …
- 20.9k
5
votes
lefschetz hyperplane section theorem
Here is an important application. It is probably in the relevant chapter of Voisins book.
Let $V^k \subset CP^n$ be a smooth projective variety, $d$ a number and $s$ be a section of the $d$th power o …
- 20.9k
3
votes
A good vector field to calculate the Euler's number of a compact differentiable manifold
This is not quite what you asked for, but hopefully similar enough. If $M^m \subset \mathbb{R}^n$ is a closed submanifold, then for almost all $v \in \mathbb{R}^n$, the restriction of the function $x …
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3
votes
Topology of maps between fibers of vector bundles
Let $G_i$, $i=0,1$, be topological groups and $P_i \to X_i$ be $G_i$-principal bundles. Then $P_0 \times P_1 \to X_0 \times X_1$ is a $G_0 \times G_1$-principal bundle. Let $V_i$ be topological vector …
- 20.9k
2
votes
Accepted
Is the space of smooth partitions of unity connected? Simply-connected?
Typically, partitions of unity are used to prove a statement along the following lines. Given a paracompact $X$ and for each open set $U\subset U$ a certain space $S_U$ which satisfies an appropriate …
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