All Questions
Tagged with dg.differential-geometry at.algebraic-topology
639 questions
5
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0
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135
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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
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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
0
votes
1
answer
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
19
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1
answer
989
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Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
- 293
5
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1
answer
371
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Orientation reversal and restriction to submanifold of lower dimension
Let $M$ be a connected closed oriented manifold with at least one orientation-reversing homeomorphism $M\to M$.
Let $S\subset M$ be a connected closed embedded submanifold of lower dimension. Let $f:M\...
- 59
0
votes
0
answers
179
views
Adjunction formula for non compact surfaces
Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$.
I already know how to show the following equality of fiber bundle:
$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...
- 25
0
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0
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170
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Cohomology ring of 5-manifold generated in degrees 1, 2, 3
Is there a connected closed 5-manifold $M$ such that $\oplus_{i\geq 0} H^i(M, \mathbb{Z})$ is generated by $H^1(M, \mathbb{Z})\oplus H^2(M, \mathbb{Z})\oplus H^3(M, \mathbb{Z})$ but is not generated ...
- 21
-1
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1
answer
137
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Manifold for which you need to specify the action on cohomology in each degree
Let $M$ be a connected closed manifold of dimension $n\geq 2$.
Can it happen that for any $I\subsetneq I_n=\{1, \dots, n\}$ there are continuous maps $f, g:M\to M$ such that $f^*=g^*|_{\oplus_{i\in I}...
- 21
1
vote
1
answer
107
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When given degrees generate the cohomology ring of a manifold?
Let $M$ be a connected closed manifold of dimension $n$.
Suppose we have a subset $I\subset I_n=\{1, \dots, n\}$ such that for any two continuous maps $f, g:M\to M$ if $f^*=g^*|_{\oplus_{i\in I}H^i(M, ...
- 21
8
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0
answers
251
views
(Higher) flat connections and Grothendieck construction
For any (nice) topological space there is an equivalence between covering spaces and local systems $\pi_1X \to \operatorname{Set}$. We can think of it as a Grothendieck construction. If we take ...
- 834
1
vote
0
answers
151
views
Density of $G$-invariant morse functions
Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
- 99
8
votes
3
answers
914
views
Spectral sequences in algebraic topology [duplicate]
What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...
4
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0
answers
158
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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
1
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240
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Question about the proof of Gromov's theorem in geodesic flows
I am trying to understand the following theorem from the book Geodesic flows :
Given a metric $g$ on a simply connected manifold $X$, there exists a constant $C_1>0$ such that given any pair of ...
user174565
6
votes
2
answers
636
views
On the Euler number of an orbit space
Let $X$ be a differentiable manifold and $G$ a finite group acting
differentiably on $X$. The following formula for the Euler number $\text{e}(X/G)$ of the orbit space $X/G$ appears to be well-known:
\...
- 1,020
7
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0
answers
350
views
Do smooth maps with nowhere-maximal rank have small image?
I’m trying to better understand the concept of “maps with small image” as used by Lipyanskiy in his construction of “geometric homology” in https://arxiv.org/abs/1409.1121. Lipyanskiy utilizes ...
- 5,489
15
votes
6
answers
2k
views
Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those ...
- 1,013
3
votes
1
answer
485
views
How to compute the index of a vectorfield defined by analytic formula?
An analytic local map (or map germ) $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0) $ can be considered as a vector field with zero at the origin. Assume that the origin is an isolated zero of $f$. How ...
- 194
15
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1
answer
1k
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Atiyah's proof of the moduli space of SD irreducible YM connections
In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a ...
- 355
3
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0
answers
137
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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
1
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0
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293
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How to calculate the total chern classes of CP^n [closed]
When calculating the total chern class of $\ CP^n$, we use the fact that their is a exact sequence of vector bundles over $\ CP^n$:
$$\ 0\to S \to C^{n+1} \to Q \to 0$$
And identify the bundle $\ TCP^...
- 157
1
vote
1
answer
453
views
Relationship between $H^1(X, \mathbb{T})$ and complex line bundles
Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, ...
13
votes
2
answers
2k
views
Given a complex vector bundle with rank higher than 1, is there always a line bundle embedded in it?
When I read the GTM 082, "The Splitting Principle of the complex vector bundle", I see that in the proof we split off one subbundle at a time by pulling back to the projectivization of a ...
- 157
3
votes
0
answers
194
views
The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$
$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...
- 2,147
9
votes
2
answers
405
views
Differential refinement of homology
Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism ...
- 5,824
9
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1
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372
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A strong form of Mostow rigidity without geometrization?
Mostow rigidity theorem says that two closed hyperbolic manifolds with isomorphic fundamental groups are isometric.
Here is my question: suppose that $M$ and $N$ are two closed 3-manifolds such that $...
- 855
7
votes
1
answer
363
views
Inverse Hodge and inverse Betti problems for Kähler manifolds
A Betti sequence is a map $\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$.
A Betti sequence $b$ is realizable if there is a connected closed Kähler manifold $M$ such that $b(k)=b_k(M)$.
A Hodge diamond ...
user164740
3
votes
1
answer
257
views
Deformation equivalent Hodge structures
An HH type is the oriented homotopy type of a closed simply-connected Kähler manifold together with the Hodge structure on cohomology.
Two HH types are deformation equivalent if they are represented ...
user164740
5
votes
1
answer
394
views
Arbitrary torsion in cohomology of Kähler manifolds
Fix an integer $i\geq 3$ and a finite abelian group $G$.
Is there a connected closed Kähler manifold $M$ such that $H^i(M, \mathbb{Z})\approx \mathbb{Z}^n\oplus G$ for some integer $n\geq 0$?
user164740
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
5
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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
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
12
votes
2
answers
767
views
Unique almost complex structure up to diffeomorphism
For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
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
1
answer
124
views
Homotoping diffeomorphism to a $J$-holomorphic one
Let $M$ be a closed simply-connected smooth manifold. Assume $M$ admits at least one almost complex structure.
Is any diffeomorphism $M\to M$ homotopic as a continuous map to a $J$-holomorphic ...
user164740
6
votes
1
answer
463
views
Antiholomorphic involution with a fixed point
Let $M$ be a connected closed complex manifold. Assume it has an antiholomorphic involution. Must it have an antiholomorphic involution with a fixed point?
user164740
18
votes
0
answers
496
views
Orientation-reversing homotopy equivalence
If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?
It is not true, for instance, that if there ...
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
13
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0
answers
319
views
Exotic smooth structures on Fano manifolds
If two Fano projective manifolds are homeomorphic are they diffeomorphic?
There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user164740
4
votes
1
answer
378
views
Chern/Hodge numbers of the conjugate complex manifold
Let $(M, J)$ be a complex projective manifold. Can $(M, -J)$ have different Chern/Hodge numbers?
user164740
5
votes
1
answer
241
views
Fano manifold becoming general type upon conjugation
Let $(M, J)$ be a Fano projective manifold. Can $(M, -J)$ be general type?
For complex curves and surfaces Kodaira dimension is diffeomorphism invariant so this cannot happen.
user164740
2
votes
1
answer
652
views
Complex manifold defined over $\mathbb{R}$
Let $M$ be a connected closed complex manifold with an antiholomorphic involution.
Must there be an atlas and a choice of a reference point in each chart such that the transition functions are ratios ...
user164740
4
votes
1
answer
447
views
Complex projective manifold with an antiholomorphic involution
Let $M$ be a complex projective manifold with an antiholomorphic involution. Can $M$ be defined by equations with real coefficients then?
user164740
9
votes
2
answers
753
views
Deformation equivalent vs diffeomorphic to projective manifold
Let $M$ be a closed complex manifold that is not deformation equivalent to a complex projective manifold.
Can $M$ be orientedly diffeomorphic to a complex projective manifold? What if $M$ is moreover ...
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
2
votes
1
answer
128
views
Infinitely many deformation equivalent Hodge diamonds
Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An ...
user164740
10
votes
2
answers
499
views
Symplectic structure on the square of a 3-manifold
Let $M$ be a connected closed orientable 3-manifold. Assume $M$ is not the direct product of a surface and the circle.
Can there be a symplectic or Kähler manifold homeomorphic to $M\times M$? I think ...
user164740
7
votes
1
answer
634
views
Hodge diamonds of complex threefolds
There is no closed complex curve or surface with $h^{1, 0}-h^{0, 1}=1$.
Now consider threefolds. Can this condition be satisfied?
Is Serre duality in fact the only restriction on the Hodge diamond?
user164740
11
votes
1
answer
379
views
Smooth structure on direct product
Let $M$ be the $E_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
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