# Tag Info

## New answers tagged algebraic-topology

1 vote

### Sections of a quotient vector bundle

Since you're working in the smooth category, every vector bundle $E$ splits as $F \oplus F^\perp$ where $F \subseteq E$ is a subbundle and $F^\perp$ its orthogonal complement in some choice of bundle ...
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### What are some interesting applications of the theory of covering spaces?

Here's some cool things you can do with covering spaces. One is any covering space of a Lie group can be again made into a Lie group. Some naturally arising Lie groups have nontrivial fundamental ...
1 vote

### Covering space action

No such counterexample exists, as a corollary of my discussion here - if Iām not wrong, of course. The key point is that local path connectivity shows its face in only one of the directions of the ...
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### Fundamental Group of the Torus Using Covering Spaces

This is the correct idea but you should mention hypotheses such as the most crucial one: $\Bbb R^2$ is simply connected. For a full discussion of hypotheses for theorems of this ilk, see here: but we ...
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### What are some interesting applications of the theory of covering spaces?

So you probably know that covering spaces can be used to prove that subgroups of free groups are free, using the fact that covering spaces of graphs are graphs. What is less well-known is that you can ...
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### Injective Resolutions in $\mathfrak{Ab}(X)$

Here's a partial answer, which also explains the "mistake" in the solution mentionned in the comments, in case someone else stumbles onto this issue like I did. The document uses a flasque ...
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### Sheaf cohomology and singular cohomology with real coefficients

In the case of $\Bbb Z$ coefficients, you seem to need exactly the same hypotheses as you would for $\Bbb R$ coefficients, or any Abelian group for that matter. Namely, the hypothesis which can be ...
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### Fundamental group of a compact Riemannian manifold whose universal covering is not compact

Shockingly, as far as I can tell this appears to be an open problem. First of all, this is a purely group-theoretic question: the fundamental group of a compact manifold is finitely presented, and ...
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### Is a homotopy long exact sequence for a pair always isomorphic to a long exact sequence coming from a fibration

The answer is no, and not even in "nice" cases. One obstruction is that a fibration comes not just with a long exact sequence in homotopy groups, but also with an (albeit more complicated) ...
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### Does weak equivalence determine the homotopy classes of maps to a CW complex?

Consider any space $X$ that is totally path-disconnected but not discrete. Then taking $\Gamma X$ to be $X$ with the discrete topology, the identity function $f:\Gamma X\to X$ is a CW approximation. ...
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### Does weak equivalence determine the homotopy classes of maps to a CW complex?

This is false. Let $X$ be the Warsaw circle, i.e. $X$ is the the topologist's sine curve in the plane together with its limit points and a choice of arc connecting the origin to 1 not intersecting the ...
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### Is there a closed 3-manifold with infinite fundamental group and trivial first homology?

The Seifert-Weber dodecahedral space is a good example, not only a homology sphere but also one of the first known closed hyperbolic manifolds, and hence has infinite fundamental group. It is also ...
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### Degree 1 maps from real projective spaces

This is a very interesting question, and (remarkably) we can decide it entirely using only relatively basic tools from the algebraic topologist's tool case. To be precise, we're going to prove the ...
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### Triviality of induced maps on (reduced) homology implies nullhomotopy?

This is false. There exist non-contractible acyclic spaces, that is, spaces whose reduced homology vanishes but which are not contractible. Such a space must have nontrivial $\pi_1$. Wikipedia offers ...
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### Is there a closed 3-manifold with infinite fundamental group and trivial first homology?

Some searching around turned up the following: the Brieskorn spheres $\Sigma(p, q, r)$ are known to be integral homology $3$-spheres if $p, q, r$ are relatively prime. Their fundamental groups were ...
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### Homology of torus with circles identified

There was some confusion about what the problem is actually asking, but after all $H_2(X) \cong \mathbb{Z}$, $H_1(X) \cong \mathbb{Z}^2$ is correct. The following argument is essentially the same as ...
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### Can open sets in $ā^n$ be finitely triangulated?

No, this is not possible (addressing your title question); indeed, every triangulation of a nonempty open subset of $\mathbb R^n$ is a infinite triangulation. For the proof, every space which has a ...
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### Homotopy groups of pointed spaces

The authors consider pointed spaces. 2.7.1 Convention. From here on, we shall be concerned mainly with pointed spaces and pointed maps. We shall use the notation $M_ā(X, Y)$ for the set of pointed ...
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### Does Dowker's theorem actually hold on bipartite graphs?

These simplicial complexes are both contractible, so they do have the same homology groups. In particular, since the "loop" you see in $K_x$ is filled in it doesn't yield a nontrivial ...
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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 $\dim e_{\sigma'} = \dim e_\sigma - k$.
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### 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 ...
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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 $Y\... • 4,029 1 vote ### Van Kampen's Theorem and simplicial complexes Assume$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 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 ... • 4,029 3 votes Accepted ### 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 ... • 438k 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 ... • 438k 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. ...
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