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9 votes
Accepted

Can we view higher homotopy groups as symmetries?

All of the higher homotopy groups are also fundamental groups, so your construction with the universal cover also applies to them. Namely, $$\pi_2(X) \cong \pi_1(\Omega X)$$ where $\Omega X$ is the ...
Qiaochu Yuan's user avatar
6 votes
Accepted

determine whether the fundamental group of a Mobius band glued to a torus is abelian

To show that a group is not abelian, it suffices to find a nonabelian quotient, eg by adding relations. Now there are many obvious quotients of the group you found that are easily seen to be ...
ronno's user avatar
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3 votes
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quotient groupoid as the colimit in the infinite category Grpd

(Homotopy) colimits of diagrams of groupoids are computed by the Thomason homotopy colimit theorem as the Grothendieck construction of the diagram. In the case under consideration, the diagram must be ...
Dmitri P.'s user avatar
  • 2,326
3 votes

When will "Retract $\iff$ Deformation Retract" hold true?

Statement 1 is almost never true. For example if $A$ is a point it is always a retract of $X$ (via the unique map $X \to A$) but if it's a deformation retract then $X$ must be contractible. Being a ...
Qiaochu Yuan's user avatar
3 votes
Accepted

Explain the use of Mayer-Vietoris in the computation of the singular homology of $S^1$

This comes down to understanding how induced maps on $H_0({{-}})$ work in general and how induced maps in general behave with respect to disjoint unions. Recall that, given any path-connected space $X$...
Ben Steffan's user avatar
  • 5,048
2 votes
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Min-Product Semiring

You probably have figured out the answer by now, but for the sake of completeness: Because both the operations should be well defined for all elements. If $0$ is an element of the set, then the ...
Enigma's user avatar
  • 36
2 votes
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When is $\pi_{k}(X_{p}^{\wedge})$ not isomorphic to the p-completion of the group $\pi_{k}(X)$?

Sure. In general the homotopy groups of $X_p^\wedge$ sit in a short exact sequence $$ 0 \to \operatorname{Ext}(\mathbb{Z} / p^\infty, \pi_* X) \to \pi_* X_p^\wedge \to \operatorname{Hom}(\mathbb{Z}...
Ben Steffan's user avatar
  • 5,048
2 votes

About the definition of universal covering space

Most authors define a universal covering as a simply connected covering space of a path-connected locally path-connected space $X$. Note that the requirement that $X$ be path-connected can be omitted. ...
Paul Frost's user avatar
  • 78.2k
2 votes

How to prove the continuity of the path lifting function of covering spaces.

If you know that $p$ is a fibration, this is follows from the exponentional law, i.e. a map $f\colon X\to E^I$ is continuous iff the adjoint map $\hat f\colon X\times I\to E, \hat f(x,t)=f(x)(t)$ is ...
Vincent Boelens's user avatar
2 votes
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Proof of the Euler characteristic of the real Grassmannian $\mathbf{G}(k, n)$

If $\sigma_1 \ne 1$, then $\sigma' = (\sigma_1 - 1, \dots, \sigma_k - 1)$ exactly runs over the symbols for the Schubert cells of $\mathrm{G}(k, n - 1)$ but $\dim e_{\sigma'} = \dim e_\sigma - k$.
ronno's user avatar
  • 12.5k
1 vote

Van Kampen's Theorem and simplicial complexes

Assume $K$ connected, otherwise pass to a connected component. Then the $1$-skeleton $G$ is connected, so pick a spanning tree $T$. Then $G \simeq G/T$ is a wedge of circles indexed by edges of $G$ ...
ronno's user avatar
  • 12.5k
1 vote
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Mapping cone homotopy equivalent to quotient Space

This is true if $f$ is a (Hurewicz) cofibration. This uses two facts. First, the pushout of a cofibration is again a cofibration, i.e. if $f\colon X\to Y$ is a cofibration and $P$ is the pushout of $Y\...
Vincent Boelens's user avatar
1 vote
Accepted

Doubt on topological proof of the fundamental theorem of algebra

Indeed, pointwise convergence $h_t(z) \to z^n$ isn't enough. What would suffice is uniform convergence: for any $\epsilon>0$ there is some $t_0$ such that $|h_t(z) - z^n| < \epsilon$ for all $z \...
arkeet's user avatar
  • 7,824

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