Questions tagged [differential-topology]
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”.
1,897 questions
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Fundamental class from smooth atlas
Let $M$ be an orientable $n$-dimensional manifold with a smooth oriented atlas $A :=\{(U_\alpha,\Psi_\alpha)\}_{\alpha}$ so that every non-empty $U_\alpha\cap U_\beta$ is contractible.
Then we can ...
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2
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1
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The definition of irreducible representation for U(n) and the corresponding vector bundle
I'm reading this paper (Endomorphism Valued Cohomology
and Gauge-Neutral Matter) and I'm not sure if I understand the notation below, quoted from the paper:
"The complex projective space $\mathbb{...
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8
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1
answer
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Finite covers of manifolds with universal cover $ S^{n-1} \times \mathbb{R} $
Let $M^n$ be a closed smooth $n$-manifold with $n \ge 3$. Suppose its universal cover is diffeomorphic to $\widetilde{M} \cong S^{n-1} \times \mathbb{R},$ where $S^{n-1}$ carries the standard smooth ...
- 2,291
3
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1
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Geometric insights from the computation of algebraic-topological invariants
(This is a sibling question of the "inverse" implication)
First, I see differential topology as a bridge between more "geometrical" data/structures/methods and more "...
- 560
0
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1
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Is every submanifold the zero of a sufficiently regular section of a vector bundle on some neighborhood of the submanifold?
Let $M$ be an $m$-manifold and $N\subseteq M$ an (embedded) $n$-submanifold $(0<n<m)$. Everything in this question is assumed smooth if relevant and not stated explicitly otherwise.
Let $\pi:E\...
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0
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187
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The Hantzsche-Wendt manifold
Does the direct product of the Hantzsche–Wendt manifold and a circle admit a complex structure?
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3
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How to generalize the notion of distance between sections to open manifolds as done for h-principles?
In Introduction to the h-principle by Cieliebak, Eliashberg and Mishachev (so the second edition), the authors state on page 26 the holonomic approximation theorem. This states that (I'm shortening ...
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A generalized Pontryagin-Thom theorem
Here is a quetion that arise naturaly from homotopical perspective on Morse-Bott theory (for example Côté and Kartal paper "Equivariant Floer homotopy via Morse-Bott theory").
The paper ...
- 560
10
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1
answer
564
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First homology groups and the universal cover
Let $M^{2n+1}$ ($n \geq 3$) be a closed Riemannian manifold. If its Riemannian universal cover is conformally equivalent to $(\mathbb{H}^2 \times S^{2n-1},\ g_{\mathbb{H}} \oplus g_{\mathrm{st}})$, ...
- 2,291
2
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0
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Hybrid of a flat vector bundle and a line bundle
I have a simple queation:
Suppose $V$ is a vector bundle that is isomorphic to a tensor product of a flat vector bundle and a line bundle $V=F\otimes L$. Where $F$ is a flat vector bundle and $L$ is ...
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4
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Examples of differential topology methods yielding new insights in algebraic topology
In differential topology, one can start with a manifold and use differential geometry calculations to obtain algebraic-topological invariants like the Euler characteristic, (co)homology groups, ...
3
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0
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Is there a Compact Non-Smoothable $4$-manifold that Topologically Immerses in $\mathbb{R}^5$?
Suppose that $M$ is a compact, connected topological $4$-manifold that has a topological immersion, i.e. local topological embedding, into $\mathbb{R}^5$. Then is $M$ necessarily smoothable? Note ...
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1
vote
1
answer
202
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An exact sequence of jet space associated to a vector bundle
Suppose $L$ be a line bundle over Riemann surface $X$. Then show that $ 0 \longrightarrow J^2(L) \longrightarrow J^1(J^1(L)) \longrightarrow L\otimes K_X \longrightarrow 0 ,$ where $J^k(L)$ is the $k$-...
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Must diffeomorphisms on $S^n$ have fixed points?
Question.
Let $N_1 \simeq S^k$ and $N_2 \simeq S^l$ be disjoint smoothly embedded spheres in $S^n$ with $k + l = n$.
Suppose a diffeomorphism $\psi: S^n \to S^n $ preserves $N_1$ and $N_2$, and that ...
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3
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0
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Explicit homotopy equivalence $\operatorname{BDiff} \rightarrow \operatorname{BGL}$
Let $\operatorname{BGL}(d)$ and $\operatorname{BDiff}(\mathbb{R}^d)$ be the simplicial Spaces defined as the nerves of the obvious topological groupoids.
I am looking for an explicit weak homotopy ...
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