# Tagged Questions

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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### 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 ...
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### 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. ...
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Original question (see also the revised, possibly simpler, version below): Let $g > 1, r > 1$ be integers. Playing around with the Verlinde formula (see below), I came across the expression $$\... 3answers 528 views ### Does every bijection of \mathbb{Z}^2 extend to a homeomorphism of \mathbb{R}^2? Given a bijection f\colon \mathbb{Z}^2 \to \mathbb{Z}^2, does there always exist a homeomorphism h\colon\mathbb{R}^2\to\mathbb{R}^2 that agrees with f on \mathbb{Z}^2? I don't see any ... 6answers 723 views ### Is the complement of countably many disjoint closed disks path connected? Let \{D_n\}_{n=1}^\infty be a family of pairwise disjoint closed disks in \mathbb{R}^2. Is the complement$$ \mathbb{R}^2 -\bigcup_{n=1}^\infty D_n $$always path connected? Here “... 4answers 3k views ### Why is the Jordan Curve Theorem not “obvious”? I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ... 2answers 974 views ### The “Easiest” non-smoothable manifold In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ... 1answer 475 views ### Decomposition of a manifold As a kind of aside to this question, where one of the answers assumed that if S^n=X \times Y then we can assume that X and Y are manifolds. If we have a manifold M, such that M is ... 4answers 886 views ### Useful fibrations What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and S^1\to S^{... 1answer 364 views ### The kernel of free group map to surface group G is a surface group of genus g\geq 2 (the fundamental group of closed orientable surface of genus g). F is a free group of rank 2g with basis \{x_1,\dots,x_{2g}\}. \phi is a surjective ... 2answers 302 views ### What closed 3-manifolds have fundamental group \Bbb Z? For certain small groups, it is easy (and desirable) to classify closed (and orientable if necessary) 3-manifolds with that group as their fundamental group. (Essentially due to Waldhausen is that for ... 1answer 500 views ### Can one cancel \mathbb R in a bi-Lipschitz embedding? Let X be a metric space. Suppose that the product X\times\mathbb R admits a bi-Lipschitz embedding into \mathbb R^{n}. Does it follow that X admits a bi-Lipschitz embedding into \mathbb R^{n-... 2answers 380 views ### How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds? Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ... 2answers 903 views ### A simply-connected closed surface is a sphere From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that S^2 is the only closed surface with trivial \pi _1. That's easy because the fundamental group ... 1answer 277 views ### Intuition for Exotic \mathbb R^4's Today one of my professors told me that \mathbb R^4 admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like ... 3answers 466 views ### Is every self-homeomorphism homotopic to a diffeomorphism? Given a smooth manifold M, is every homeomorphism M \to M homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every C^1 diffeomorphism of smooth manifolds is ... 1answer 390 views ### Is the torus the union of two connected, simply-connected open sets? Is the torus the union of two connected, simply-connected open sets? A routine computation with the Mayer-Vietoris sequence shows that if so, then their intersection must have exactly three components.... 4answers 2k views ### Reference on Geometric Topology Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional ... 1answer 324 views ### Maps from D^n to D^n with a single inverse set are open. Let D^n denote the closed unit ball in \Bbb R^n. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ... 1answer 683 views ### Gluing a solid torus to a solid torus with annulus inside. I was thinking the fact that if two genus 1 handlebodies (solid tori) are glued via an orientation preserving homeomorphism of boundaries, the resulting manifold depends only on (up to isotopy) ... 1answer 1k views ### Why is the knot group of the trefoil isomorphic to the group of 3-braids? I apologise in advance for the vagueness of this question but I have not been able to find very much info on the topic and have made very little progress on my own. I am trying to understand why the ... 1answer 426 views ### Equivalence of Definitions of Principal G-bundle I've finally gotten around to learning about principal G-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ... 1answer 287 views ### A quadratic reciprocity formula Inspired by a problem of calculating explicitly the invariants by Reshetikhin and Turaev for certain 3-manifolds, I have come across the following problem involving Gauss sums: I would like to prove ... 1answer 178 views ### Equivalence of definitions of S^\infty Consider the following two definitions of the infinity-sphere S^\infty. Why do they define homeomorphic spaces? 1) The set of points in \mathbb R^\infty with distance 1 from the origin. 2) ... 1answer 404 views ### Space of homeomorphisms Homeo(S^1) of S^1 deformation retracts onto O(2) How can we prove that the space of homeomorphisms Homeo(S^1) of S^1 (strong) deformation retracts onto the orthogonal group O(2)? I know that this result is proved by Hellmuth Kneser in his ... 1answer 217 views ### Is the E8 manifold homeomorphic to a CW complex? Is the E8 manifold homeomorphic to a CW complex? (I know that it is not triangulable) Edit: The E8 manifold is the unique compact (without boundary), simply connected topological 4-manifold, whose ... 1answer 505 views ### Covering points on a sphere with a disk Suppose m points ("sites") are selected on the unit sphere S^2. For a given radius r < \pi, we can define a disk around any point on the sphere as the set of points at geodesic distance at ... 1answer 390 views ### Fractional versions of euclidean space? This is going to be a somewhat vague question, but I'll be happy if you indulge me. Euclidean space \mathbb{R}^n is equipped with a lot of nice (algebraic, metric, topological,...) structure and ... 4answers 937 views ### Embedding compact (boundaryless?) n-manifolds in n-dimensional real space I know the embedding theorems that allow you to embed n-manifolds into \mathbb{R}^k, provided k is chosen large enough. Here I'm interested in the possibility of taking k=n in the case of ... 2answers 4k views ### Understanding the Equation of a Möbius Strip I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a math-... 2answers 273 views ### Which groups act freely on S^n? When n is even, it is easy to classify groups which act freely on S^n using degree theory: if G acts on S^n, then associating to each element g \in G the degree of the map obtained from ... 1answer 171 views ### General relationship between braid groups and mapping class groups I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ... 2answers 410 views ### \{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k} The question is motivated by the notion of handle attachment, Morse theory, critical points of index k, Morse lemma, sublevel sets, etc. For 0\!\leq\!k\!\leq\!n and 0\!<\!\varepsilon\!<\!1... 1answer 280 views ### Nontrivial h-cobordism I'm learning the h-cobordism theorem as I want to use it in a talk. I'd like to be able to give an example of an h-cobordism that isn't a cylinder, if possible by drawing a picture. What is the ... 2answers 190 views ### Partitioning the plane into three sets each intersecting the vertices of every square with side 1? Q1. Is it possible to partition the plane into three sets such that each of them contains at least one vertex of every square with side 1 ? (I mean all squares of side-length 1, not just those with ... 2answers 619 views ### Does the Euler characteristic of a manifold depend upon the field of coefficients? Define the Euler characteristic of a space X to be$$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$This is obviously not necessarily well-defined for an arbitrary space X, so let X be a manifold (... 1answer 95 views ### Two disjoint real projective planes in real projective space? Let \mathbb{R}\mathbb{P}^3 be the real projective three-space. It is clear that any two hyperplanes in \mathbb{R}\mathbb{P}^3 intersect. But I wonder whether one could embed two copies of the real ... 1answer 612 views ### Spin structures on S^1 and Spin cobordism I'm trying to understand the 2 spin structures on the circle. Since the frame bundle for the circle is just the circle itself, Spin structures on S^1 correspond to double covers of S^1. There are ... 1answer 122 views ### What are the 8 non-compact Euclidean 3-manifolds? I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but ... 1answer 608 views ### Easier proof about suspension of a manifold For what manifolds M is the suspension \Sigma M also a manifold? By the suspension of a topological space X (not necessarily a manifold), I mean the space$$\Sigma X = (X \times [0,1])/{\sim} ...
There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ ...