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      All Questions

      1
      vote
      0answers
      213 views

      On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$

      I am confused with the following argument. I know I am doing something wrong but I can't find my mistake. On one hand, one knows that if $M$ is a Lie group, then $$\mathrm{Diff}(M)\simeq M\times\...
      2
      votes
      0answers
      88 views

      Space of non-vanshing sections path-connected?

      Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...
      6
      votes
      0answers
      243 views

      Quotient space, a fundamental group, and higher homotopy groups 2

      Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
      2
      votes
      0answers
      254 views

      Quotient space, homogeneous space, and higher homotopy groups

      Preparation and my input: For the quotient space $G/H$, knowing the homotopy groups of $G$ and $H$ one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(H) \to \pi_n(G) ...
      5
      votes
      0answers
      176 views

      Asphericity of hypersurface complement in ${\mathbb C}^n$

      How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$. One way I can think of is to give ...
      5
      votes
      0answers
      65 views

      Bounding the dimension of the euclidean space in which any $n$-manifold embeds “$k$-uniquely” in it

      (The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary). I'm interested in the ...
      2
      votes
      1answer
      255 views

      Are there compact flat fiber bundles with “truly” infinite structure group?

      Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds. I am looking for a counterexample for the following statement: $E \cong \widetilde{B}...
      5
      votes
      0answers
      252 views

      What is the local structure of a fibration?

      It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled. Here I'd like ...
      19
      votes
      2answers
      1k views

      A manifold is a homotopy type and _what_ extra structure?

      Motivation: Surfaces Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy ...
      3
      votes
      0answers
      106 views

      Weak contractibility of some infinite dimensional metric spaces

      Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: $X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
      9
      votes
      4answers
      601 views

      Complements of Simply Connected Subsets of the Plane

      this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...
      21
      votes
      2answers
      422 views

      Morphism from a surface group to a symmetric group, lifted to the braid group

      Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
      2
      votes
      1answer
      133 views

      Classification of pretzel links up to link homotopy using alexander quandle

      I am currently reading this paper where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}_{k}[t^{\pm 1}]\diagup_{(t-1)^{2}}$, and I find it ...
      3
      votes
      1answer
      255 views

      Linking number a complete invariant of link homotopy

      I read in Milnor's article "Link groups", where he defines invariants to classify links up to link homotopy, that the linking number is a complete invariant which can tell almost trivial two ...
      7
      votes
      1answer
      390 views

      Isomorphism from $\mathbb{Z}$ to third homotopy group of compact simple Lie group

      Let $G$ be a compact connected simple Lie group. It is known that its third homotopy group $\pi_3(G)$ is isomorphic to $\mathbb{Z}$. More precisely, there is a Lie group homomorphism $$\rho:SU(2)\...

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