# Tagged Questions

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### fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
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### Presentations of fundamental groups regarding cones of simplicial subcomplexes

Let $L$ be a simplicial subcomplex of $K$. Let $CL$ be the cone on L. Let $X = CL \bigcup K$. Show that X is a simplicial complex and dscribe a presentation for $\pi_1(|X|,v)$ in terms of ...
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### Fundamental Group of Punctured Plane

What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is ...
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### Edges and Vertices in relation to Free subgroups

Let $\Gamma$ be a finite connected graph (1-dimensional simplicial complex), with $V ( \Gamma)$ vertices and $E(\Gamma )$ edges (1-simplices). Show that $\pi_1(\Gamma, v)$ is a free group with ...
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### fundamental group of complex numbers?

Let $\mathbb{C}^*=\mathbb{C}-{0}$. What is the fundamental group $\mathbb{C}/G,$ where G is the group of homeomorphism $\{\phi^n ; n\in \mathbb{Z}\}$ with $\phi(z)=2z$? I think the fundamental group ...
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### Homotopy equivalence -> element in outer automorphism of fundamental group

Let $X$ be a path-connected topological space. for $x \in X, G = \pi_1(X,x)$ Show that a homotopy equivalence $f : X \to X$ gives a well-defi ned element $g \in Out(G)$. How might one begin on this ...
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### Fundamental group: Attaching a disc to a path connected triangulable space

What's the effect on the fundamental group of a path connected triangulable space X if you attach a disc? So far I've figured out that a disc has trivial fundamental group. So the free product of ...
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### a problem about surface $M_{g}$.

1.$M_{g}$ has normal universal cover $\widetilde{X}$ with deck transformation $G(\widetilde{X})=\mathbb{Z}^{n}$ if and only if $n \leq 2g$. 2.for $n=3,g \geq 3$ explain such covering. 3.show that ...
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### if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,also $A\subset X$ is a path connected subset,show that $p^{-1}(A)$ is path connected. I suppose that ...
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### find a necessery and enough condition just using $\pi_{1}$.

suppose $p:\widetilde{X} \rightarrow X$ will be a covering space and $X$ is path connected and locally path connected, also $\widetilde{X}$ is connected, then find a necessary and enough condition ...
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### $H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1

Let $A,B$ be two open set of a topological space, $H_1(A\cap B)\to H_1B$ is 1-1 implies $H_1A\to H_1(A\cup B)$ is 1-1, where the homomorphisms is induced by inclusion. I feel that using the ...
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### Show that the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n},0,0)$ is simply-connected.

Show that the subspace of $\mathbb{R}^{3}$ that is the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n},0,0)$ for $n=1,2,3,...$ is simply-connected. for showing it is ...
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### showing that $\Phi:\Pi_{1}(X,x_{0})\rightarrow [S^{1},X]$ is onto if $X$ is path connected.

We can regard $\Pi_{1}(X,x_{0})$ as the set of basepoint-preserving homotopy classes of maps $(S^{1},s_{0})\rightarrow(X,x_{0})$ . Let $[S^{1},X]$ be the set of homotopy classes of maps ...
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### Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ? And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )?? Please help me. Thank you
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### If the function $\varphi \colon Z\rightarrow C(X,Y)$ is continuous then $F\colon Z\times X\rightarrow Y$, $F(z,x)=\varphi (z)(x)$ will be continuous.

If the function $\varphi :Z\rightarrow C(X,Y)$ ($C(X,Y)$ with compact-open topology) is continuous and $X$ is locally compact, then $$F\colon Z\times X\rightarrow Y$$ $$F(z,x)=\varphi (z)(x)$$ will be ...
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### Determining the generators of cohomology (as a ring)

I am working on a problem to show that the cohomology graded rings of $\mathbb{C}P^3$ and $S^2$ x $S^4$ are not isomorphic (unreduced with integer coefficients) I have already calculated the graded ...
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### Find Group Action

I'm asked to find a group action G on the unit cylinder C such that $C/G$ is homeomorphic to the torus. Would $\pm1 \cdot (x,y,z) = (x,y,\pm z)$ work? The only problem here is that G should only act ...
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### Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
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### Homology of connected sum of real projective spaces

Let $A_k=RP^2\sharp RP^2\sharp \cdots \sharp RP^2$ be a connected sum of $k$ copies of real projective space. With coefficients in $\mathbb{Z}$, it is clear $H_n(A_k)=0$ when $n\geq2$ and ...
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### Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let ...
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### Is $S^1, S^1 \times S^1, S^1 \vee S^1$ closed?

There seems some more obvious reasons for that $S^1, S^1 \times S^1, S^1 \vee S^1$ closed? I am thinking of seeing $S^1,S^1 \vee S^1$ as the subcomplex of some CW complex, such as $S^2$, and seeing ...
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### $D^2 \cup \{pt\} = S^2$?

I think if I can ignore the metric, then $D^2$ only differs $S^2$ by a point, namely, the infinity. But I am wondering that if it is true that $D^2 \cup \{pt\} = S^2$? Thank you
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### Question on relative homology

i have this: where $|\tau|$ is the support of the chain $\tau$, i don't understand the first part why $[\sigma]=0$ in $H_{m-1}(\phi^{c+\varepsilon},\emptyset)$ ??? Please, thank you.
I don't understand why I can't connect the $-1$ and $1$ points with just two line segments. I've tried it in my head and it makes sense to me. Why do I need $3$ line segments? Can somebody draw this ...