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Results tagged with differential-topology
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user 8103
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”.
34
votes
Accepted
Is differential topology a dying field?
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
- 35.9k
22
votes
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
Edit: The answer below is incorrect. In fact, $\bar{w}_3(\mathbb{R}P^2\times\mathbb{R}P^2)=0$ (thanks to Rafal Walczak for pointing this out) so by the cited result $\mathbb{R}P^2\times\mathbb{R}P^2$ …
- 35.9k
19
votes
Accepted
homology classes as immersed submanifolds
It might be better to split the question into 2 cases and 2 steps.
Step 1: Which homology classes in $X$ can be represented by continuous maps of closed smooth manifolds (ie, which classes are $f_*[M …
- 35.9k
16
votes
Accepted
Classification of $O(2)$-bundles in terms of characteristic classes
The $O(2)$ bundles $\xi$ over a manifold $M$ are classified by their first Stiefel-Whitney class $w_1(\xi)\in H^1(M;\mathbb{Z}/2)$ and their twisted Euler class $e(\xi)\in H^2(M;\mathbb{Z}_{w_1(\xi)}) …
- 35.9k
16
votes
Can the nth projective space be covered by n charts?
It seems worth giving the cup-length argument, as it's relatively short and sweet.
Suppose $\mathbb{R}P^n=U_1\cup\cdots\cup U_n$, with each $U_i\approx\mathbb{R}^n$, and let $c\in H^1(\mathbb{R}P^n;\m …
- 35.9k
16
votes
Accepted
Non orientable, closed manifold covered by two contractible charts
No. Recall that the Lusternik-Schnirelmann category of a space $X$, denoted $\operatorname{cat}(X)$, is the minumum $k$ such that $X$ may be covered by open sets $U_0,U_1,\ldots, U_k$ such that each i …
- 35.9k
14
votes
Integral homology classes that can be represented by immersed submanifolds but not embedded ...
This is a great question, and I don't have an answer but this is too long for a comment.
Working mod $2$, a codimension $k$ homology class $z\in H_{m-k}(M;\mathbb{Z}/2)$ is realizable by an embedding …
- 35.9k
13
votes
Does every vector bundle allow a finite trivialization cover?
This should be a comment to the answer of Andreas Blass, but was too long.
The Lusternik-Schnirelmann category $\operatorname{cat}(X)$ of a space $X$ is the smallest number $k$ such that $X$ has a co …
- 35.9k
12
votes
Accepted
Homotopy type of $SO(4)/SO(2)$
This is the Stiefel manifold $V_2(\mathbb{R}^4)$. It fits into a fibration $S^2\to V_2(\mathbb{R}^4)\to S^3$ and so has trivial $\pi_1$. By playing around with the long exact sequence of this fibratio …
- 35.9k
11
votes
Can a Morse function define a unique structure on a closed manifold?
In looking for counter-examples it helps to notice that $M$ and $N$ have CW structures with the same numbers of $i$ cells, and hence they have the same Euler characteristic.
It turns out that there i …
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11
votes
Accepted
Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Very roughly speaking: Geodesics $\gamma:[0,1]\to M$ with $\gamma(0)=\gamma(1)=p$ and $\operatorname{ind}(\gamma)<k$ correspond to critical points of Morse index less than $k$ of the energy functional …
- 35.9k
10
votes
Accepted
Ambient isotopy of the diagonal submanifold in product space
The answer is always no for $M$ a closed manifold.
If $\Delta: M\to M\times M$ were isotopic into the first factor, then in particular it would be homotopic to a map $\Delta': M\to M\times M$ with i …
- 35.9k
10
votes
Book recommendation for cobordism theory
The book Differentiable Periodic Maps by Conner and Floyd is a classic reference. Despite its age, the book is very easy to understand and gives clear expositions of some important topics which are di …
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9
votes
Euler characteristic and universal cover
Here is sketch proof that $\chi(\tilde{M})=0$ implies that $\chi(M)=0$ for fundamental groups with certain finiteness properties.
The idea is to adapt the usual spectral sequence proof that the Euler …
- 35.9k
9
votes
Accepted
Integral homology classes of which no multiples admit embedded representatives with trivial ...
With the trivial normal bundle condition, it's fairly easy to produce non-realizable examples using Théorème II.2 of Thom's paper. Namely, a class $z\in H_l(M^n;\mathbb{Z})$ is realizable by an embedd …
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