The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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78
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0answers
5k views

Grothendieck 's question - any update?

I was reading Barry Mazur's biography and come across this part: Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first ...
31
votes
0answers
893 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
8
votes
0answers
268 views

When are maps between topological manifolds automatically surjective?

Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective. I have ...
7
votes
0answers
68 views

Is “slightly deform” a well defined concept in mathematical proof?

In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept. ...
6
votes
0answers
166 views

Symmetric product of genus 2-surface

Let $\Sigma$ be the genus 2-surface. Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$. In the very ...
6
votes
0answers
96 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
5
votes
0answers
61 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
5
votes
0answers
47 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
5
votes
0answers
131 views

$\operatorname{Spin}^c(n)$ is a Lie group?

Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.) Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
5
votes
0answers
82 views

Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?

I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
5
votes
0answers
150 views

Heegaard Splitting of Brieskorn homology 3-spheres

For pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. I want to know ...
5
votes
0answers
243 views

Points in the plane at integer distances

Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds: For all $a,b$ with ...
4
votes
0answers
55 views

Surgery or Partial Whitney Trick in McDuff and Salamon proof of Gromov squeezing

In the proof of Gromov squeezing in McDuff and Salamon's epic J-Holomorphic Curves and Symplectic Topology, the authors use a surgery argument without any explicit construction. In the following, ...
4
votes
0answers
60 views

What is a 2-surgery on a disk?

I am confused by a certain point in Scharlemann's paper "Sutured Manifolds and Generalized Thurston Norms", which seems important enough to not just skip it. I mean the "2-surgery on disks" in the ...
4
votes
0answers
406 views

The Birman–Hilden Theorem and the Nielsen–Thurston classification

So this post is half question/half reference request, as I'm sure it's the kind of thing people would have thought about before (and indeed the question might even be trivial), but I've been unable to ...
4
votes
0answers
404 views

Soft question: why are there non-smooth manifolds?

Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...
4
votes
0answers
119 views

Problem from Hempel — surfaces in Lens spaces

This isn't a homework problem, just working through Hempel's 3-Manifolds for my own benefit. One exercise is to show that the lens space $L(2k,q)$ contains a surface of Euler characteristic $2-k$. ...
4
votes
0answers
247 views

Partitioning a triangulated 2-sphere into two triangulated discs

Take a triangulation of the 2-sphere, $S^2$. Let the triangulation be denoted $T$. The Euler characteristic tells you that the number of triangles in $T$ is even. Since triangulations of the ...
4
votes
0answers
152 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...
4
votes
0answers
71 views

Are PL maps determined by PL-paths?

Let $X,Y$ be polyhedra. If $f\colon X\rightarrow Y$ is a PL map and $g\colon I\rightarrow X$ is a PL map (where $I$ denotes the interval), then $f\circ g$ is a PL map. Is the converse true? i.e. ...
3
votes
0answers
34 views

Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
3
votes
0answers
73 views

Homotopically equivalent manifolds and product with $\mathbb{R}$

I know that some manifolds which are homotopically equivalent become homeomorphic after taking the product with $\mathbb{R}$, e.g. $\mathbb{T}^{2}$ minus a point and $\mathbb{S}^{2}$ minus three ...
3
votes
0answers
51 views

What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
3
votes
0answers
143 views

Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
3
votes
0answers
51 views

References on the relations between Top, Diff and PL

I have heard many times informal statements like "differentiable and pl manifolds are essentially the same for such and such dimensions", but I would like to know what they mean exactly and how such ...
3
votes
0answers
61 views

Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
3
votes
0answers
71 views

Actual Definition of the term: Hopf Band?

Sorry if this is too trivial: I need an actual working definition of the term: Hopf band. I see references to it in many searches, but never an actual precise definition. All I know so far is that ...
3
votes
0answers
37 views

Are there closed manifolds that can be given multiple geometric structures of constant curvature

For example, is there a closed differentiable manifold that can be given both a Euclidean structure and a Hyperbolic structure? If not, is there a reasonably easy proof that no such manifold exists?
3
votes
0answers
80 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
3
votes
0answers
165 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
3
votes
0answers
100 views

What is the “Standard” Open Book Decomposition for $\mathbb R^n$, and why does this matter?

I am trying to understand better Open book decompositions. To that effect, I tried to work out a couple of (relatively-simple) examples, specifically, for $\mathbb R^2 $ and higher. But I have not ...
3
votes
0answers
81 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
3
votes
0answers
39 views

What is the “real osculating space” of an immersion?

In a differential geometry paper from 1979 I have come across some terminology which I have not found explained anywhere else. We have an immersion $x : S^2 \to S^n$. In the paper, it is a minimal ...
3
votes
0answers
202 views

Fundamental Group!

Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
3
votes
0answers
198 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
3
votes
0answers
120 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
3
votes
0answers
134 views

Something like the weak Whitney embedding theorem for continuous maps and homotopy.

This is sort of a reference request. Consider a continuous map of orientable topological manifolds $f:N\longrightarrow M$ of dimension $n$ and $m$ respectively. I have been told that there is a ...
2
votes
0answers
35 views

Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...
2
votes
0answers
22 views

Euler Integral of a self-overlapping tube with a cusp singularity

I am studying in depth the following paper on Euler calculus applied to target enumeration: https://www.math.upenn.edu/~ghrist/preprints/eulerenumerationpart1.pdf Within this paper there is an ...
2
votes
0answers
40 views

cohomology ring of cross-section space of fibre-bundles

Given an $m$-dimensional manifold $M$, let $TM$ be the tangent bundle of $M$ and $SM$ be the $m$-sphere bundle over $M$ obtained by fibre-wise one point compactification of $TM$. Let $\Gamma(SM)$ be ...
2
votes
0answers
66 views

Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
2
votes
0answers
26 views

Linear vs smooth actions of finite groups on spheres, Euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following: (1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ but admitting no effective linear action on ...
2
votes
0answers
26 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
2
votes
0answers
25 views

pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...
2
votes
0answers
82 views

Which spaces admit bump functions?

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets. Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...
2
votes
0answers
60 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
2
votes
0answers
67 views

A homotopy equivalence between two sets

I was trying to prove that the set consisting of the union of the circles $\{\langle x,y\rangle\mid(x-10)^2 +y^2 = 1\}$, $\{\langle x,y\rangle\mid(x+10)^2 +y^2 = 1\}$ and line segment $\{\langle x, ...
2
votes
0answers
32 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
2
votes
0answers
37 views

Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
2
votes
0answers
32 views

Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...