All Questions
Tagged with dg.differential-geometry at.algebraic-topology
172 questions with no upvoted or accepted answers
3
votes
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answers
121
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Topology of abstract varieties over $\mathbb{C}$
What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think
they have to have non-zero $b_2$ ...
user132250
3
votes
0
answers
98
views
Euler characteristic of an exhaustion of compacts of a surface
Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$
...
- 750
3
votes
0
answers
75
views
Two questions regarding flat fibre bundles and the corresponding group action on the fibre
Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...
- 404
3
votes
0
answers
337
views
Quotient space of Grassmannian
The Grassmannian $G(k,2k)$ is equipped with a nice $\mathbb Z_2$ action with respect to a non-degenerate symplectic bilinear form: $1.V=V^{perp}$. Is there a reference where the ring of polynomial ...
- 673
3
votes
0
answers
270
views
Fiber bundle in smooth category and topological category
Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundles on $M$ with structural group $G$ in smooth category. And denote by $Bun(M,G)...
- 31
3
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490
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Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$
I'll first state the question as concisely as I can and then provide some motivation.
Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
- 21.5k
3
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0
answers
446
views
When does the normal bundle of a submanifold of Euclidean space admit a flat connection?
Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the ...
- 313
2
votes
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answers
116
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Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
- 131
2
votes
0
answers
101
views
A roof genus of high dimensional lens space
Let $p$ be a natural number, and for $i\in \{0,
..., p-1\}$,
denote the irreducible rank one complex representation of $\mathbb{Z}/p$. by $\rho_{i}$.
Let $a=(a_{1},\ldots a_{d}) $ ...
- 2,630
2
votes
0
answers
208
views
Classification of bundles with fixed total space
I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
- 501
2
votes
0
answers
175
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Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
- 447
2
votes
0
answers
241
views
Monodromy group action on de Rham cohomology
Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
- 2,907
2
votes
0
answers
137
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Question about spin map
I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
- 563
2
votes
0
answers
58
views
Dimension changes from global to local immersion
From Hatcher Corollary A.10. the (global) immersion for an $n-$dimensional CW complex is possible in some $\mathbb{R}^N$. I have started with $M(G,n)$ (Moore space of type $(G,n)$, $G$ is cyclic ...
- 1,177
2
votes
0
answers
168
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Geometric sets determined by chains (for integration and Stokes' theorem)
I have asked a similar question on mathSE more than a year ago, which received no answers, only a few comments which did not really help me. I am now re-asking this question here but reformulated ...
- 2,792
2
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0
answers
214
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Kollar-Matsusaka's finiteness theorem from a topological perspective
I am looking to find a reference or proof for the following topological version of Kollar-Matsusaka's theorem, which does not seem to be stated explicitly in the literature.
First I recall the ...
- 6,995
2
votes
0
answers
485
views
Splitting principle for real vector bundles
I'm reading the Book of John Roe, Elliptic Operators, Topology and Asymptotic Methods and got stuck at Lemma 2.27.
i) How does this lemma show that a real vector bundle can be given by a pullback of ...
- 145
2
votes
0
answers
152
views
When are automorphisms of the cohomology ring realized by isometries?
Let $(M,g)$ be a closed smooth Riemannian manifold, and denote by $G$ a closed subgroup of its isometry group. By considering the maps $g^*$ induced by elements $g\in G$ in the (de Rham) cohomology $H^...
- 5,891
2
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0
answers
243
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What does one call a Morse function whose nondegenerate condition is relaxed?
In robotics, navigation functions are of utmost interest to plan a path from an initial location $q_0$ to a target location $q^t$. A function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a navigation ...
user106458
2
votes
0
answers
305
views
Are homotopy equivalent manifolds with homeomorpic boundaries themselves homeomorphic?
Let $f:M \to M′$ be a homotopy equivalence of topological manifolds with boundary such that $dim(M)=dim(M′)$ and $f:\partial M \to \partial M′$ is a homeomorphism. Does this imply the existence of a ...
- 79
2
votes
0
answers
235
views
Volume form on real Grassmanian
There exists a unique $SO(4)-$invariant volume form on the real Grassmanian G(2,2). Is it possible to write it down algebraically in Plucker coordinates?
- 4,316
2
votes
0
answers
161
views
Chern character (form) of a Gauss-Manin connection
Consider the trivial fibration $\mathbb{T}^2\to\mathbb{S}^1$, where $\mathbb{T}^2$ is the two-torus. Denote by $\mathbb{C}\to\mathbb{T}^2$ the trivial line bundle over $\mathbb{T}^2$, and equip it ...
- 1,173
2
votes
0
answers
105
views
Multiplicativity of the analytic index (or of kernel bundle)
What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators.
In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
- 1,173
2
votes
0
answers
188
views
How many linear independent vector fields can be constructed on a general manifold with $\chi(M)=0$?
We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres
So how many linear independent vector fields can be ...
- 977
2
votes
0
answers
385
views
Intuition behind the following theorem of Reeb?
What is the intuition behind the following theorem of Reeb?
If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.
- 29
2
votes
0
answers
429
views
A Generalized De Rham cohomology
Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by $L_{\mathbb{C}}^{...
- 356
2
votes
0
answers
189
views
About the Lie algebra of polyvector fields
I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
- 1,609
2
votes
0
answers
652
views
Manifold with nonzero pontryagin number?
I am a physics student, for some physical property, I am seeking a manifold with nonzero pontryagin number. And I was told CP(2) is the simplest example which has $p_1(CP(2))=3$.
Is there a nice(...
- 575
2
votes
0
answers
179
views
About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)
Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...
- 1,100
2
votes
0
answers
409
views
virtual bundle with compact support
A virtual bundle with compact support is a triple $E$={$E_1$,$E_2$,$a$},
where $E_1$ and $E_2$ are bundles over $M$ of the same dimension, $M$ is a closed manifold, $a$ is a bundle map from $E_1$ to $...
- 381
1
vote
0
answers
133
views
Stable equivalence and stability theorem of vector bundles
I am going through this paper by Tanaka. In the proof of Proposition 3.2(1) given below
The author says that by the stability theorem as $\dim (B)\le m$ we have $\alpha\oplus1\cong m\oplus1$. But I ...
1
vote
0
answers
153
views
Poincaré-Hopf Theorem for domains with a point of vanishing curvature
Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes.
I would like to know ...
- 111
1
vote
0
answers
151
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Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper,
he wrote in eq.2.22 that
Atiyah Patodi Singer theorem says that the combination:
$$
\frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi}
$$
is a ...
- 447
1
vote
0
answers
155
views
Lifting action of torus to torus bundle
Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...
- 205
1
vote
0
answers
64
views
Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
1
vote
0
answers
185
views
Proving a Result About Pontryagin Numbers Without Forms
I've been reading the book Geometry of Differential Forms by Shigeyuki Morita, and I came across the following theorem on page 226 the other day:
Proposition 5.53 (Pontryagin). Two cobordant closed (...
- 377
1
vote
0
answers
167
views
Spectral sequence for two fibrations
Given maps of fibrations, i.e. commutative diagrams of smooth manifolds
$$\begin{matrix}
\ F & \to & E &\to & B \\\
\downarrow & & \downarrow & & \downarrow \\\
\ F'...
- 103
1
vote
0
answers
97
views
about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
1
vote
0
answers
108
views
Positive $(p,p)$-current and subvariety
Let $X$ be a complex manifold. Any dimension $p$ subvariety(or reduced analytic subscheme) will determine a real closed positive $(p,p)$-current. But the converse is not true.
My question:
Is there a ...
- 361
1
vote
0
answers
246
views
Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle
Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...
- 1,915
1
vote
0
answers
157
views
Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,527
1
vote
0
answers
32
views
Results on compact slices in a regular foliation
Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...
- 133
1
vote
0
answers
284
views
A question on existence of gradient vector field on manifold with boundary
Let $M$ be a compact manifold with smooth boundary $\partial M$. Does $M$ admit a gradient vector field $\nabla u$, which has no zeros, i.e. $\nabla u(x)\neq 0$, $\forall x\in M\cup\partial M$?
Thanks ...
- 51
1
vote
0
answers
194
views
Existence of Morse function on suspension
Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...
- 61
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
1
vote
0
answers
137
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Covers of a 4-manifold pull back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
Is ...
user164740
1
vote
0
answers
116
views
Defining the cospecialization in topology
Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...
- 10.5k
1
vote
0
answers
300
views
Constructions that can be seen as objects representing a functor
Some constructions can be seen as objects representing a functor.
For example,
Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...
- 6,023
1
vote
0
answers
150
views
Nontrivial Gauss-Manin connection
Suppose $p: X \rightarrow S$ is a fiber bundle of smooth manifold, if the Gauss-Manin connection is nontrivial, could $p$ be trivial bundle as smooth manifold? Also, could $p$ be trivial bundle as ...
- 677
1
vote
0
answers
57
views
$\omega$-nilpotent cover of a recurrent surface
Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville.
$\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
- 391