# Tag Info

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 ...
• 433k
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 ...
• 12.5k
Accepted

### 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 ...
• 2,326

### 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 ...
• 433k
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 $X$...
• 5,048
Accepted

### 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 ...
• 36
Accepted

### 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}...
• 5,048

### 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. ...
• 78.2k

### 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 ...
• 3,969
Accepted

### 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 $\dim e_{\sigma'} = \dim e_\sigma - k$.
• 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 $G$ ...
• 12.5k
1 vote
Accepted

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 $Y\... • 3,969 1 vote Accepted ### Doubt on topological proof of the fundamental theorem of algebra Indeed, pointwise convergence$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 \...
• 7,824

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