All Questions
Tagged with dg.differential-geometry at.algebraic-topology
172 questions with no upvoted or accepted answers
6
votes
0
answers
397
views
Differential forms on the simplex which are "constant towards the boundary"
Let $\Delta^k$ the standard $k$-dimensional simplex, $\Delta^k=\{(x^0,\dots,x^k)\in \mathbb{R}^{k+1}|\sum_{i=0}^kx^i=1\}$, and let $\Omega^\bullet(\Delta^k)$ be the de Rham complex of smooth ...
- 6,777
5
votes
0
answers
233
views
Classification of principal $\mathrm{SO}(3)$-bundles on a 4-manifold via characteristic classes
I am interested in a reference with a detailed (as simple and topological as possible) proof of the following fact:
Theorem. A principal $\mathrm{SO}(3)$-bundle on a compact oriented 4-manifold are ...
- 6,962
5
votes
0
answers
179
views
Deformations of cotangent bundles
Let $M$ be a smooth variety of even dimension over $\mathbb C$. I am interested in necessary or sufficient conditions such that $X$ is a deformation of a family of cotangent bundles.
In other words, ...
- 6,622
5
votes
0
answers
248
views
Algebraic de Rham cohomology with torus coefficients
Let $X$ be a smooth projective variety over $\mathbb{C}.$
On page 3 in this preprint of Simpson, it is stated that
Notice first of all that the algebraic de Rham theory is not going to work well in ...
- 123
5
votes
0
answers
121
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How to see $\delta_2(\hat{\chi}(V))=\chi(V)$ in differential cohomology?
I'm reading the paper "Differential Characters and Geometric Invariants" by Cheeger and Simons. In Page 62 the authors defined the differential Euler character $\hat{\chi}(V)\in \hat{H}^{2n-...
- 9,019
5
votes
0
answers
133
views
On the vertical cohomology of a fibered manifold
Let $\pi:Y\rightarrow X$ be a $C^\infty$ fibered manifold (all constructions, unless otherwise stated are over the smooth manifold category) with $\Omega_k$ the sheaf of (smooth) $k$-forms on $Y$.
...
- 2,792
5
votes
0
answers
280
views
Was an index theorem for manifold with local boundary condition proven?
I would like to ask a question on the bibliography of the index theorems on manifold with boundary.
Before my bibliographical research my understanding of the field was that for manifold with boundary,...
- 51
5
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0
answers
132
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 4,081
5
votes
0
answers
135
views
Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
5
votes
0
answers
120
views
How does the topology of minimal surfaces depend on the radius?
Let $M^n \subset \mathbf{R}^{n+k}$ be a smooth, properly embedded minimal surface, with boundary $\partial M$. The convex hull property states that $M$ is contained inside the convex hull of its ...
- 5,038
5
votes
0
answers
130
views
Minimal sum of Betti numbers of Kähler manifold with trivial canonical bundle
Let $M$ be a closed Kähler manifold of real dimension $2n$. Suppose the canonical bundle of $M$ is holomorphically trivial.
Is it true that $\sum_{i=0}^{2n} b_i(M)=n+3\implies n=1$?
user164740
5
votes
0
answers
297
views
Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
- 2,247
5
votes
0
answers
140
views
Reference request: Name or use of this group of diffeomorphisms of the disc
Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...
- 5,405
5
votes
0
answers
543
views
a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"
I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71:
$$H_{2n-q}(S)...
- 1,121
5
votes
0
answers
238
views
Tensor product of "difference bundles" ( Atiyah construction)
There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor ...
- 51
5
votes
0
answers
167
views
h-principle on Hilbert manifolds
Gromov's h-principle is a powerful tool in studying various geometric structure on open, finite-dimensional manifolds. Is there any generalization of h-principle to (necessarily open) infinite-...
- 606
5
votes
0
answers
135
views
Smoothing a continuous section in 1-jet bundle
Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle).
I ...
- 51
5
votes
0
answers
604
views
The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)
(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
- 10.5k
5
votes
0
answers
717
views
Can Euler Class be defined by the Splitting Principle for Real Vector Bundles?
Let $M$ be a manifold and $S$ its sphere bundle with fiber $\mathbb{S}^n$. As we know, the notion of the Euler class is raised from the problem of finding a global form on $S$ which restricts on each ...
- 643
4
votes
0
answers
177
views
Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
- 141
4
votes
0
answers
196
views
Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?
Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...
- 1,379
4
votes
0
answers
113
views
Is there a general definition of twisted Real equivariant cohomology theory?
There are some classical examples of Real equivariant cohomology theories and twisted cohomology theories, including equivariant KR-theory in Atiyah and Segal's paper, and the more general ...
- 1,040
4
votes
0
answers
181
views
Specify the embedding of special unitary group in a Spin group via their representation map
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
4
votes
0
answers
158
views
Postnikov square explicitly on a simplicial complex
$\DeclareMathOperator\Z{\mathbb{Z}}$
Following Wikipedia, a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by ...
- 10.5k
4
votes
0
answers
138
views
Topology of Brody hyperbolic manifolds
Let $M$ be a Brody hyperbolic complex projective manifold with $\pi_1(M)=\{0\}$. Can $M$ be homeomorphic to $P\times S^2$ where $P$ is a manifold?
user164740
4
votes
0
answers
273
views
Instanton numbers for diverse gauge bundles on diverse manifolds --- their relations to characteristic classes
It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$
$$
n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
- 10.5k
4
votes
0
answers
294
views
Various definitions of the odd Chern character form
I am asking this question from my possibly defected memory, so the things below may not be accurate.
I want to know how many different definitions of the odd Chern character form using differential ...
- 1,173
4
votes
0
answers
293
views
Cohomology of the classifying space of some Super Lie group
Are there any papers on the cohomology of the classifying space of the general linear supergroup $GL(n, m)$ or unitary supergroup $U(n, m)$?
I know basically nothing about supergeometry. It seems ...
- 1,173
4
votes
0
answers
302
views
Fundamental groups of stably parallelizable manifolds
Is it possible to realize every finitely presented solvable group as a fundamental group of a stably parallelizable closed n-manifold? If not, are there any known counterexamples?
4
votes
0
answers
362
views
Weil Kostant Integrality Result as Stated by Brylisnki
I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued 2-...
- 1,611
4
votes
1
answer
370
views
Compatible reductions of the structure group of a principal fiber bundle
Let $P$ be a principal bundle over a manifold $M$ with structure group the Lie group $G$. Assume that $P$ admits to distinct topological reductions, say $Q_{1}$ and $Q_{2}$, where $Q_{a}$, $a=1,2$ are ...
- 2,816
3
votes
0
answers
360
views
on definitions of stacks
There are two ways to define a stack.
The first one is that the presheaf of sets Isom (a,b) is a sheaf and that every descent data is effective.
The second one says that a stack is a homotopy sheaf of ...
- 494
3
votes
0
answers
195
views
Is there such an isotopy for every homology sphere?
Let $n \geq 3$, and $\Sigma^{n-1} \subset \mathbf{S}^n$ be a smoothly and properly embedded, orientable, and connected submanifold of the sphere. This divides the sphere into two open sets, $U_-$ and $...
- 5,038
3
votes
0
answers
137
views
Intersection number for 4 manifold with boundary
Let $X$ be a closed oriented smooth $4$-manifold. Suppose there is an embedding $\Sigma\to X$, it is known that the self-intersection number satisfies $[\Sigma]\cdot [\Sigma]=\pm\int_\Sigma c_1(N)$, ...
- 1,915
3
votes
0
answers
40
views
Non-existence of local generators for Sobolev tangent subundles
Let $U\subset\mathbb R^n$ be a bounded open set, a rank $r\le n$ measurable tangent subbundle $\mathcal V$ on $U$ is a map to the Grassmannian $\mathcal V:U\to Gr(r,\mathbb R^n)$ which is only defined ...
- 938
3
votes
0
answers
137
views
On the construction of principal $S^1$-bundles with prescribed characteristic form
I am trying to understand an argument made by Kobayashi (https://projecteuclid.org/download/pdf_1/euclid.tmj/1178245006, page 35) in his construction of a principal $S^1$-bundle with connection $1$-...
- 1,227
3
votes
0
answers
194
views
The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$
$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...
- 2,147
3
votes
0
answers
55
views
Infinitely many deformation equivalent Hodge diamonds II
Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
...
user164740
3
votes
0
answers
86
views
Exotic smooth structures on HK manifolds
An HK manifold is a closed simply-connected Kähler manifold $M$ such that $H^0(M, \Omega_M^2)=\mathbb{C}\omega$, where $\omega$ is a holomorphic 2-form on $M$ which is nowhere degenerate as a skew-...
user164740
3
votes
0
answers
71
views
Holomorphic homeomorphisms
Let $M$ be a connected closed smooth manifold. Consider the group $\mathrm{Homeo}(M)$ of homeomorphisms $M\to M$ endowed with the $C^0$-topology.
If $M$ has a symplectic structure some people study ...
user164740
3
votes
0
answers
98
views
Non-diffeomorphic surface bundles over homeomorphic 4-manifolds
For a smooth manifold $M$ an $M$-surface is the total space of a smooth surface bundle over $M$.
Let $M_1$ and $M_2$ be two homeomorphic closed simply-connected smooth 4-manifolds. Can there be an $...
user164740
3
votes
0
answers
162
views
Exotic smooth structures on Calabi-Yau manifolds
A Calabi-Yau manifold is a simply-connected closed Kähler manifold with holomorphically trivial canonical bundle and $h^{2, 0}=0$.
If two Calabi-Yau manifolds are homeomorphic are they diffeomorphic?
user164740
3
votes
0
answers
109
views
Kähler manifolds deformation equivalent to projective manifolds
Let $M$ be a closed non-projective Kähler manifold. There are three possibilities
there is a proper holomorphic submersion $f:X\to \Delta$ with $f^{-1}(0)\cong M$ such that the projective fibers ...
user164740
3
votes
0
answers
144
views
All Kähler threefolds embed into a common complex manifold
Is there a closed complex manifold into which all closed complex threefolds admitting a Kähler structure embed?
user164740
3
votes
0
answers
147
views
Index bounded Riemannian metrics
Let $L$ be a closed simply-connected smooth manifold with a Riemannian metric $g$. We say $g$ is index bounded if the energy functional (which is assumed to be Morse/Morse-Bott)
$$
E: C^k(L,g) \...
- 41
3
votes
0
answers
119
views
Factorizing vector fields near manifolds of singularities
Let $V: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth vector field containing a smooth $k$-dimensional manifold $M$ (with $1\leq k < n$) of singularities: $V(M)=0$.
Suppose furthermore that at every ...
- 105
3
votes
0
answers
95
views
Decomposing a compact connected Lie group
I want to prove the following. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ and center $Z_G.$ It is not hard to prove that $\mathfrak g$ is reductive. Therefore, we can ...
- 1,879
3
votes
0
answers
186
views
Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
- 10.5k
3
votes
0
answers
166
views
geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold
It's my first post.
Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...
- 31
3
votes
0
answers
127
views
Methods for constructing or checking for nontrivial classes in de Rham cohomology with local coefficients
Let $M$ be a smooth manifold (possibly with boundary), $E \to M$ a flat vector bundle, and $\mathcal{L}$ the corresponding sheaf of parallel sections.
Given a de Rham cohomology class $[\omega] \in H^...
- 6,025