Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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
vote
0answers
41 views

Can the rank of harmonic maps decrease far from the boundary?

This is a cross-post. Let $\mathbb D^n$ be the closed unit disk in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a smooth immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be the ...
4
votes
1answer
71 views

perturbing one map to be transverse to a second map

Let $f\colon M \to N$ and $g\colon A \to N$ be two smooth maps between manifolds $A,M,N$. Can one perturb $f$ to be transverse to $g$ (without touching $g$)? Transverse meaning: For every $y\in f(M)\...
10
votes
0answers
131 views

Are there exotic twisted doubles of 4-manifolds?

Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...
5
votes
1answer
208 views

Conversion formula between “generalized” Stiefel-Whitney class of real vector bundles: O(n) and SO(n)

$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$, $$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$ Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as: $$ w_j(...
0
votes
1answer
60 views

Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$

Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$. Consider the de Rham complex $(\Omega^*(M), d)$ and the cochain complex $$ C:=\mathrm{Hom} (\...
2
votes
1answer
82 views

regularity of harmonic forms on manifolds-with-boundaries

Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
0
votes
0answers
27 views

limit derivative convex function exist

Let $U$ be an open convex subset of $\mathbb{R}^{n}$ and $f(y,t)$ are real continuous convex function on $U$. We assume that $x_{n}\rightarrow x$ and $f(x_{n},t_{n})$ is diffrentaible with respect to $...
1
vote
0answers
86 views

Do orientation preserving diffeomorphisms preserve homological intersection?

Let $M$ be a $4$- dimensional oriented manifold and let D be a 2-dimensional submanifold. Is it true that any $\phi \in \text{Diff}^+(M)$(the set of orientation preserving diffeomorphisms) preserves ...
2
votes
1answer
142 views

Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...
2
votes
0answers
116 views

Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$?

Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing $$ H^1(X) \times \pi_1(X) \to \...
2
votes
1answer
281 views

Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$. I ...
3
votes
1answer
204 views

Homotopy type of $SO(4)/SO(2)$

A classical result states that the quotient $SO(4)/SO(3)$ is homotopy equivalent to $S^3$. In fact, this can be stated in more general terms since $SO(n+1)/SO(n)$ has the homotopy type of $S^n$. What ...
5
votes
1answer
310 views

On limits of manifolds

This question should be fairly elementary. I’d just like to check I’m not missing anything. Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...
2
votes
1answer
95 views

Integrability of an almost complex structure vs holomorphicity of the section $M\rightarrow \mathcal{J}(M)$

Let's say we have an almost complex manifold $(M, J)$. Consider the complex vector bundle $V\rightarrow M$ whose fiber over $x$ is the space of almost complex structures on $T_x M$. Is there any ...
6
votes
1answer
191 views

Action of diffeomorphism group on non-vanishing vector fields

Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...
5
votes
1answer
193 views

Characteristic classes of the bundle of trace free, skew adjoint endomorphisms

In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}_E$ of trace-free, skew adjoint automorphisms of $...
11
votes
0answers
106 views

Intrepreting Spin(3) as a certain configuration space

Let $\mathcal{C}$ denote the space of great circles in $\mathbb{S}^2\subset \mathbb{R}^3$. It's pretty easy to see that any element $\mathcal{C}$ can be identified uniquely with the axial line (in $\...
3
votes
0answers
71 views

Generic properties of Jacobians of smooth functions

Let $f = (f_1, \dotso, f_n):\mathbb{R}^n \to \mathbb{R}^n$ be a smooth map and let $J$ be its Jacobian (determinant of the matrix with $ij$-th entry $\partial_i f_j$). We introduce the zero sets of $J$...
14
votes
0answers
207 views

Presenting 3-manifolds by planar graphs

From a planar graph $\Gamma$, equipped with an integer-valued weight function $d:E(\Gamma) \sqcup V(\Gamma) \to \mathbb{Z}$, one can build a $3$-manifold $M_{\Gamma}$ as follows. For each vertex $v$, ...
5
votes
0answers
85 views

Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
5
votes
0answers
277 views

Elementary questions about Morse-Bott functions

Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...
6
votes
0answers
117 views

Cell structures of simply-connected 5-manifolds (classified by Barden's 1965 paper)

In Barden's 1965 paper: Simply-connected five manifolds, Barden gave a complete list of diffeomorphism classes of simply-connected 5-manifolds: $$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$ ...
0
votes
0answers
110 views

What insight does a regular group-action of a linear algebraic group on an algebraic variety give about the isolated points of the latter?

I'm trying to gather clues on solving this bigger problem (MO link). So, say I have an algebraic variety $\Omega$ over a field $F$ (e.g $\mathbb R$, $\mathbb C$, $\mathbb F_2$, $\mathbb F_q$). ...
2
votes
0answers
273 views

Counting / characterizing the isolated points of a particular algebraic variety

I'm not a professional geometer / topologist, so please thanks for your patience :) Setup The following questions are the first in a series of steps I'm undertaking in an attempt to break down a ...
7
votes
0answers
346 views

Understanding a certain algebraic set arising in Deep Learning

I'm not a professional geometer. Thanks in advance for your patience. So, let $n$, $k$, $p_0,\ldots,p_{k}$ be positive integers. Let $X$ (resp. $Y$) be an $p_0$-by-$n$ (resp. an $p_{k}$-by-$n$) real ...
7
votes
0answers
160 views

Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
3
votes
1answer
182 views

Is there a vector field such that one differential form is the Lie derivative of the other?

I'm looking for a reference or answer for the following question: Let $M$ be an (compact and orientable, if it helps) smooth manifold and $\nu$ and $\mu$ two differential forms. I'm looking for ...
4
votes
0answers
115 views

Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer

Atiyah-Hitchin-Singer Ref 1 states that the number of virtual dimensions of the instanton moduli space for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...
2
votes
1answer
195 views

Embedding problem for 3-manifolds attacked via 4-manifolds

In this archiv paper which is continuation of following: Borodzik, Maciej; Némethi, András; Ranicki, Andrew, Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16, No. 2, 971-1023 (2016). ...
7
votes
0answers
201 views

The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons

In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
3
votes
1answer
123 views

A geometric property about certain polynomials in two variables

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...
6
votes
1answer
166 views

Do an unlinked trefoil and figure-eight cobound an annulus in $B^4$?

Let $K_1$ the trefoil (left or right hopefully does not matter?) and let $K_2$ be the figure-eight knot in $S^3 = \partial B^4$. Are there any smooth properly embedded annulus $A$ in $B^4$ with $\...
6
votes
0answers
185 views

Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?

Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and ...
7
votes
1answer
93 views

Minimum number of double points over all immersed disks

Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to ...
6
votes
1answer
162 views

Smooth functions on subsets of $\mathbb{R}^n$

I am teaching a course in basic differential topology, and, following e.g. Milnor, I defined functions of class $C^k$ on subsets of the Euclidean space $\mathbb{R}^n$ as follows. Let $f\colon X\to \...
4
votes
2answers
307 views

Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

I apologize in advance if this question has an obvious answer. Let $(M,g)$ be a Riemannian manifold. Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
8
votes
2answers
442 views

Measures and differential forms on manifolds

Let $M$ be a differentiable manifold. Let $\mu$ be a (probability) measure on $M$. What are the conditions under which $\mu$ is given by a differential form on $M$? I imagine some sort of ...
3
votes
2answers
139 views

Theorems similar to Tischler fibering theorem

Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other diffential topology ...
4
votes
1answer
113 views

Possible orders of automorphisms for the Poincare homology sphere

Let $M^3$ denote the Poincare homology sphere. I am wondering what the possible orders of (smooth) automorphisms of $M$ are (I'm not sure if allowing arbitrary homeomorphisms changes things?). By ...
0
votes
0answers
109 views

Geometric interpretation of torsion homology classes

Suppose I have a homology class $x \in H_1(M)$ which is torsion of order $k$ say. Suppose furthermore that $M$ has Dimension big enough, such that every element of $H_1$ and $H_2$ can be relalized as ...
3
votes
1answer
133 views

Computing relative cohomology class of differential form

When dealing with a top degree differential form $\mu$ in a manifold $M$, a way of "computing" its cohomology class is integrating it through the whole manifold. For instance, if the integral $ \int_M ...
3
votes
0answers
82 views

Trying to understand why this local coordinates parametrizes a manifold

First of all, I would like to say that I think this question fits better on Math Overflow than on Math Stack Exchange, in view of the proposal of the two sites. However, if my analysis of the ...
4
votes
1answer
252 views

Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure

The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
5
votes
1answer
244 views

A possible characterization of sphere or projective space

Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \...
4
votes
0answers
128 views

Separating submanifolds in $\mathbb R^n$

Let $M\subset\mathbb{R}^n$ be a submanifold. If M is diffeomorphic to $\mathbb{R}^{n-1}$ and $\mathbb{R}^n\smallsetminus M$ has two components, then must the components be diffeomorphic to $\mathbb{R}^...
2
votes
0answers
54 views

Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?

$\newcommand{\GL}{\operatorname{GL}}$ Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
4
votes
0answers
164 views

Non-spin 5-manifold and $2^2$-Bockstein homomorphism

The $2^2$-Bockstein is $\beta_4$ is associated to $$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$ (The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
0
votes
3answers
95 views

On the properness of the graph of a convex function

Let $f : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a smooth and convex function. Let us assume that $\Gamma_f = \mathrm{graph}(f) $ is a complete hypersurface of $\mathbb{R}^{n+1}$. Then I know ...
4
votes
1answer
143 views

Stable normal framings of parallelizable manifolds

Suppose $M$ is a compact, connected, orientable manifold ($\dim M=m$) with trivial tangent bundle and let $j \colon M \to \mathbb R^n$ be an embedding. Suppose we choose a trivialization of $TM$. Then ...
3
votes
1answer
141 views

Exponential map/ Lie derivative in variation for constant formula for ODE

In short: The question is how to go from the first equation on page 8, of this paper to the second equation. Some background I'm working in optimization and I am currently reading a paper see page ...