# 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 ...
• 437k
Accepted

### What is a homotopy pushout? Non-categorical terms please

Let me try to give an elementary answer. Let $A \overset{f}{\leftarrow} B \overset{g}{\rightarrow} C$ be a span of topological spaces, which is simply to say that $A, B$ and $C$ are topological spaces ...
• 5,527
Accepted

### Higher homotopy/homology groups of smash product of a space

This is pretty much hopeless for homotopy groups (at least, if you want a statement in arbitrary degrees). For instance, the homotopy groups of $S^1$ are completely known and very simple, but the ...
• 334k

### Homotopy and Homology groups of two disks glued with twists

Let me just give you a hint to get started. You're wrong off the bat; the two discs are not really discs. Note $z\mapsto z^2$ is not injective: the maps $D^2\to X$ are not embeddings, the two disc-...
• 43.3k
Accepted

### Confusing Remark on the Deformation Retraction from $\mathbf{X}$ to $\mathsf{X}?$ - Hatcher's Algebraic Topology

As defined about a page before the part you quoted, A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t : X \to X$, $t \in I$, such that $f_0 = 1$ (the identity map), ...
• 124k
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

### Can open sets in $ℝ^n$ 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 ...
• 124k
Accepted

### 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 ...
• 5,527

### 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 ...
• 124k

### Homotopy and Homology groups of two disks glued with twists

As FShrike points out in their answer, your approach does not work. While the approach using van Kampen and Mayer-Vietoris is nice, let me give a slightly different approach. To compute $\pi_1(X)$, ...
• 5,527
Accepted

### 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) ...
• 5,527