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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”.
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 …
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 …
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 …
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 …
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 …
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 …
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.
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 …
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 …
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 …
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 …
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; …
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 …
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 …
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 …