# Tagged Questions

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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### There is more then one Homology Theory for spaces, which are not Hausdorff

All of us know: if we have a CW-complex, then using an arbitrary homology theory (with the axioms of disjoint union and dimension axiom) we always get the same homology groups up to an isomorphism. ...
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### Prove that there exists a long exact sequence…

Let $f, g : X \to Y$ be two continuous maps. Consider the space $Z$ which is obtained from the disjoint union of $Y$ with $(X \times [0, 1])$ by identifying $(x, 0) \sim f (x), (x, 1) \sim g(x),$ for ...
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### Derivation and meaning of a long exact sequence of a Homotopy Groups for pairs of spaces

I read that there are long exact sequences of homotopy groups for each pair of pointed spaces $(X,A,x_{0})$. Now I know that for an exact sequence that, as the example below denotes $f \text{ and } g$...
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### Fundamental group of the complement of the solid torus

Let $T$ be a solid torus, how to calculate the fundamental group $\pi_1(\mathbb R^3- T)$? intuitively, I think it's a free group with one generator. So if it is so, how to obtain it, and what the ...
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### The fundamental group of the unit disc with one point removed from its boundary

If $y\in \partial (\mathbb D^2)$, then how to find $\pi_1(\mathbb D^2-\{y\})$? I know that if $y$ was an interior point then the answer will be $\mathbb Z$. But why both cases would be similar ?
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### Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The (...
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### $K$-theoretical interpretation of Bott periodicity.

We consider complex Bott periodicity $\pi_{i-1}(U) \simeq \pi_{i+1}(U)$. Why we can say that the $K$-theoretical interpretation of this assertion is $\tilde{K}(X) \cong \tilde{K}(\Sigma^2(X))$? Where ...
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### Definition of the fundamental group

Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What ...
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Let be $X,Y$ Hausdorff spaces. I denote with $\langle \cdot, \cdot \rangle$ the basepoint preserving homotopy classes of maps, with $\Sigma$ the reduced suspesion and with $\Omega$ the loop space. How ...
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### Why is this homotopy an isotopy?

I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states: $\eta_t(u)$ is a homeomorphism for ...
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### A step in computing the cohomology ring of $\mathbb{C}P^n$

On page 250 of Hatcher's Algebraic Topology, he uses a certain corollary to compute the cohomology ring of $\mathbb{C}P^n$. The relevant section is below for convenience: I understand the proof ...
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### The loop space of the classifying space is the group: $\Omega(BG) \simeq G$

Why does delooping the classifying space of a topological group G return a space homotopy equivalent to G. In symbols, why $\Omega(BG) \cong G$, where $G$ is a topological group and $BG$ its ...
Let $S^2$ the unit sphere. We can consider the associated path fibration $$\Omega(S^2) \rightarrow P(S^2) \rightarrow S^2 .$$ I have to explain path fibration so I think that it is useful to make a ...
### Is $\langle abab^{-1}\rangle$ a normal subgroup of $\langle a,b|\varnothing\rangle$?
Let $G=\langle a,b | \varnothing\rangle$ and let $H\leq G$ s.t $H=\langle abab^{-1}\rangle$. Is $H\triangleleft G$? I'm asking this question in order to understand the fundamental group of the Klein ...