# Tagged Questions

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”.

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31 views

### How to show $\operatorname{Int}(z)$ is not empty?

For any closed smooth curve $z:[a,b]\to \Bbb C$, we define the interior $\operatorname{Int}(z)$ of $z$ as follows:
Because $\operatorname{Im}(z)$ is bounded, we can find a circle $C$ such that $\...

**5**

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75 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 $\text{Pin}^-$...

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194 views

+50

### 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 ...

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107 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:
$$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$
...

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106 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 $\mathbb R$, $\mathbb C$, $\mathbb F_2$, $\mathbb F_q$).
...

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257 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 ...

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321 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 ...

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154 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**

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**1**answer

173 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 ...

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110 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 ...

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**1**answer

185 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). ...

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193 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 ...

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**1**answer

121 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 ...

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**1**answer

159 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 $\...

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168 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 ...

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**1**answer

90 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 ...

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**1**answer

145 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 \...

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298 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 $\omega_g$...

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402 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 ...

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**2**answers

133 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 ...

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**1**answer

108 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 ...

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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 ...

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**1**answer

118 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 ...

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80 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 ...

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**1**answer

240 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) ...

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**1**answer

221 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, \...

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125 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}^...

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50 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{...

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124 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}(-,\...

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88 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 ...

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**1**answer

131 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 ...

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**1**answer

136 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 ...

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**1**answer

143 views

### Are isotopic and conjugate homeomorphisms, conjugate by an element in $\mathrm{Homeo}_0(M)$?

An answer to this question would also answer Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
Let $M$ be a topological manifold and let $f,g$ be two orientation ...

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**1**answer

478 views

### What can we say about the Cartesian product of a manifold with its exotic copy?

Let $M$ be a smooth oriented manifold, and let $M^E$ be an exotic copy, i.e homeomorphic but not diffeomorphic to $M$.
Is it true that $M\times M$ is diffeomorphic to $M\times M^E$?
I am ...

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**0**answers

71 views

### When are two codimension-one foliations of a manifold M “diffeomorphic”?

By the question, I mean: Given two different codimension-one foliations of a manifold M, $\mathcal{F}_i$ and $\mathcal{F}_j$, when does there exist an element of Diff(M) that maps each leaf of $\...

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300 views

### Torus action implying infinite fundamental group

Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?
Consider the ...

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**1**answer

284 views

### Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...

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277 views

### Deforming a section to a section without zeros

Let $M$ be an oriented manifold of dimension $n$. Suppose furthermore that $E$ is an oriented vector bundle of rank $n-1$ over $M$. Let $s$ be a section of $E$ transversal to the zero section in $E$. ...

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45 views

### Must bounded, closed, smooth curves with long straights have sharp bends?

Consider the family of bounded, closed, and continous curves $\Gamma$, i.e. for all $\gamma \in \Gamma$, we have $\gamma : [0, 1) \to [0, 1]^2$. Within this family, I am interested in curves that ...

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112 views

### Quotient by a non-free action of a Lie group and manifolds with corners

The quotient manifold theorem says that
If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$...

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**1**answer

235 views

### Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...

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**2**answers

440 views

### Any 3-manifold can be realized as the boundary of a 4-manifold

We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...

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429 views

### What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?

The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...

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**1**answer

367 views

### Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...

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**1**answer

201 views

### Thom's first isotopy lemma

Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that
$f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...

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203 views

### Is a Difference of Fiber Bundles a Fiber Bundle?

I have a seemingly very basic question in differential topology, but I could not find the answer by a short google search.
Let $M,N$ be smooth manifolds, and let $f:M\to N$ be a smooth fiber-bundle,
...

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57 views

### Locally trivial fibration of hyperplane arrangement complement

I have a very specific question. How does one check the following map
${\mathbb C}^n-\cup \{z_i=\pm z_j\ for\ i\neq j\}\to {({\mathbb C}^*)}^{n-1}-\cup\{z_i= z_j\ for\ i\neq j\}$ defined by $(z_1,z_2,...

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**1**answer

120 views

### What is symplectic cut of a 4-ball?

Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the $\mathbb{S}^1$ orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. ...

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**2**answers

514 views

### Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold.
a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ?
b) If not, are there some concrete ...

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**1**answer

150 views

### How to calculate the Chern class of the tensor product of a torsion free sheaf with a line bundle

I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c_{r}(E\otimes L) = \sum_{...