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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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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
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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
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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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1answer
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 ...
4
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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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1answer
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 ...
3
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1answer
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 ...
6
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1answer
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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1answer
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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1answer
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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2answers
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 ...
3
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2answers
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 ...
4
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1answer
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 ...
3
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1answer
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 ...
4
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1answer
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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1answer
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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3answers
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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1answer
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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1answer
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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1answer
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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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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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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1answer
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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1answer
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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0answers
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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1answer
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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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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1answer
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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1answer
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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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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1answer
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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2answers
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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1answer
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_{...