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Results tagged with differential-topology
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user 10366
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”.
47
votes
How can there be topological 4-manifolds with no differentiable structure?
Let's try doing this with a compact, closed, connected $1$-manifold $M$. Certainly I can choose a topological embedding $f:M\to\mathbb{R}^2$. However, the image might be very fractal, like a Koch sn …
- 56.9k
30
votes
Accepted
Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ f...
I'll use cohomology with coefficients $\mathbb{Z}/2$ everywhere.
Suppose that the space $P=\mathbb{R}P^{n-1}$ embeds in $S^{n}$ (where $n>2$). Recall that
$$ H^*(P)=(\mathbb{Z}/2)[x]/x^{n} = (\math …
- 56.9k
19
votes
Accepted
On the fundamental group of a finite CW complex
Let $X$ be a CW-complex, and write $X_k$ for the $k$-skeleton. The cellular approximation theorem says that any based map $S^1\to X$ is homotopic to a cellular map, and that any two cellular maps tha …
- 56.9k
16
votes
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in ...
Bröcker and Jänich's "Introduction to Differential Topology" is admirably short and nicely written, and you may find that it contains everything that you need.
- 56.9k
13
votes
Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
I'll assume that $1\leq k<d$, the other cases being easy. Then the long exact sequence of homotopy groups shows that $B$ is simply connected, so we have a Serre spectral sequence with untwisted coeff …
- 56.9k
11
votes
Accepted
Analytic maps in homotopy classes
This is always true. The Morrey-Grauert theorem says that $M$ has a real-analytic embedding in Euclidean space, so real-analytic functions $M\to\mathbb{R}$ separate points, so they are dense in the a …
- 56.9k
10
votes
Accepted
A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)
The right framework of definitions is as follows.
You have a space $E=\text{Emb}(M,\mathbb{R}^n)$ of smooth embeddings, topologised in a way that respects all derivatives. In more detail, we give $ …
- 56.9k
10
votes
Accepted
Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite d...
You can just use the embedding $f\colon G/H=U(n)/(U(k)\times U(n-k))\to M_n(\mathbb{C})$ given by $f(gH)=gpg^{-1}$, where $p=1_k\oplus 0_{n-k}$. This gives a homeomorphism from $G/H$ to the space
$$ …
- 56.9k
8
votes
Accepted
smooth manifolds as real algebraic set (continued)
It isn't too easy to find good equations for surfaces as algebraic subspaces of $\mathbb{R}^3$, but if you are willing to use $\mathbb{R}^n$ for larger $n$ then the picture is clearer. There are stan …
- 56.9k
8
votes
Accepted
Does the projectivization of a vector bundle have sections?
I'll assume you're asking about sections of $PE$. The bundle $E$ has Chern classes $c_i(E)\in H^{2i}(X)$ and thus a Chern polynomial $f_E(t)=\sum_{i=0}^nc_{n-i}(E)t^i$, where $n=\dim(E)$. A section …
- 56.9k
8
votes
Spin bordism group of classifying space $BG$ with a finite Abelian $G$
The paper of Anderson, Brown and Peterson also shows that the localisation of the spectrum $MSpin$ splits as a wedge of suspensions of the real connective $K$-theory spectrum $kO$ and various closely …
- 56.9k
8
votes
Accepted
How to classify continuous/differentiable maps from $T^2$ to $U(N)$?
You are asking about the structure of the set $M=[T^2,U(N)]$ of homotopy classes of maps from $T^2$ to $U(N)$. The functor $H_1$ gives a map $M\to\text{Hom}(H_1(T^2),H_1(U(n)))$, and standard calcula …
- 56.9k
7
votes
Accepted
Intersection of non transverse submanifolds
Let $M$ be any manifold, and let $Z$ be a closed subset of $M$. Suppose there exists a smooth function $f:M\to\mathbb{R}$ with $f^{-1}\{0\}=Z$. We can then take $X=M\times\mathbb{R}$ and identify $M …
- 56.9k
6
votes
Treating the Connected Sum (and other constructions) as a Push-out
I don't think it is true that this kind of construction is a pushout in the smooth category. Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=|x|$. This is smooth on $(-\infty,0]$ an …
- 56.9k
6
votes
Accepted
Submersions of closed manifolds
Let $F$ be the homotopy fibre of $f$ (ie the space of pairs $(x,u)$, where $x\in M$ and $u$ is a path from $f(x)$ to a specified baspoint in $N$). If $f$ is homotopic to a submersion $f'$, then (usin …
- 56.9k