0
votes
0answers
30 views

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 $az=\overline{z}$. ...
0
votes
0answers
15 views

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 ...
1
vote
2answers
66 views

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 ...
0
votes
0answers
22 views

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 ...
1
vote
1answer
44 views

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 ...
3
votes
1answer
73 views

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 ...
2
votes
1answer
49 views

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 ...
0
votes
0answers
26 views

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 ...
0
votes
3answers
46 views

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 ...
0
votes
0answers
79 views

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 ...
2
votes
0answers
35 views

$H_1(A\cap B)\to H_1B$ 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 ...
3
votes
2answers
83 views

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 ...
2
votes
0answers
48 views

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 ...
-1
votes
1answer
49 views

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
4
votes
2answers
65 views

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 ...
0
votes
1answer
53 views

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 ...
2
votes
0answers
39 views

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 ...
1
vote
1answer
45 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
0
votes
1answer
80 views

Showing that two spaces are homotopy equivalent

Let $x_0 \in S^1 \times S^1$. I want to show that $(S^1 \times S^1) - \{x_0\}$ and $S^1 \vee S^1$ are homotopy equivalent. We have to show that $\exists$ maps $f: X \rightarrow Y$ and $g: Y ...
5
votes
1answer
165 views

Hatcher Problem 2.2.36

I am struggling with the following question (2.2.36) from Hatcher for quite some time now: Show that $H_i(X\times S^n) \simeq H_i(X) \oplus H_{i-n}(X)$. I don't know how to use the hint given by ...
1
vote
0answers
63 views

prove that after removing countable infinite point from $S^{2}$,it will remain path-connected.

prove that after removing countable infinite point from $S^{2}$,it will remain path-connected. it was the question that arise in the algebraic topology course where I have this term.I thought ...
3
votes
1answer
91 views

Why is the graph $K_{3,3} $ not one skeleton of the sphere?

I know that someone has already asked the same question here, but there is no solution for part two of the question. And I'm interested in the second part. Here the question: Suppose we build $S^2$ ...
4
votes
1answer
87 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
0
votes
1answer
43 views

Prove that $S^n$ is a strong deformation retract of $\mathbb{R} \backslash\{(0,…,0)\}$

Prove that $S^n$ (n-dimensional sphere with unit radius) is a strong deformation retract of $\mathbb{R} \backslash\{(0,...,0)\}$ This is my attempt: Consider $f:S^n \rightarrow \mathbb{R} ...
5
votes
2answers
243 views

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 ...
2
votes
1answer
67 views

Show the existence of a certain subgroup of F2

This is a homework question: I need to find three subgroups of the free group with two generators, $F_2$, with certain properties. I have found the other two by constructing covering spaces of $S^1 ...
0
votes
1answer
65 views

Homology and Exact Sequence

I have this exact sequence: $$0\stackrel{f}{\rightarrow} H_k(X,C)\stackrel{g}{\rightarrow} H_k(X,A)\stackrel{h}{\rightarrow} 0$$ Can I say that $H_k(X,A)=H_k(X,C)$ and why? Please; Thank you.
5
votes
1answer
70 views

Homotopy between two functions to a circle.

Suppose $f,g: X\to S^1$ are such that $f(x)\neq -g(x)$ for any $x\in X$. I need to construct a homotopy between these two functions. Now, the fact that $f(x)\neq -g(x)$ guarantees that there is always ...
1
vote
1answer
94 views

a Problem about Degree of Map between Spheres

When I read the book "Algebra Topology-A First Course", I find a problem. It is on the Page 97, Exercise (16.15). Problem We define $f$, $g\colon \mathbb{S}^{n}\to\mathbb{S}^{n}$ to be orthogonal ...
3
votes
1answer
59 views

Orbit space of S3/S1 is S2

I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same ...
2
votes
2answers
80 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
2
votes
1answer
55 views

mayer vietoris homework

Let $X=A\cup B\ \ A,B$ are open and $A\cap B$ is contractible. Prove that $H_i(A\cup B)\equiv H_i(A)\oplus H_i(B)$ for $i\geq 2$. I think about using Mayer Vietoris sequence but I don't know how to ...
1
vote
1answer
51 views

Abelianized fundamental group

Let $P$ be the projective plane and let $nP$ be the connected sum of $n$ copies of the projective plane. Show that the abelianized fundamental group $\pi_{1}(nP)/[\pi_1,\pi_1]$ is the direct sum of a ...
2
votes
1answer
73 views

Antipodal points of sphere

Whenever $S^2$ is the union of three closed subsets $A_1$, $A_2$, and $A_3$, then at least one of these sets must contain a pair of antipodal points {${x,-x}$} in $S^{2}$ This is homework from ...
2
votes
1answer
55 views

need help with problem on homology group

Let $A_n=\{z\in \mathbb{C}\mid z^n$ is non-negative real number$\}$ then find $H_1(A_n,A_n-\{0\})$ $H_1(A_n,A_n-\{z\})$ when $0\not=z\in A_n$ show that $A_n$ is not homeomorphic to $A_m$ when ...
3
votes
2answers
69 views

Prove these 3 spaces are homotopy equivalent

The image is below. (a) $S^2$ with a diameter. (b) $T^2$ with a disk in the middle hole. (c) $S^2$ tangent with $S^1$ . I think they may the deformation retract of the same space. But I can't ...
0
votes
0answers
42 views

Prove on $S^1$ deg(f)=deg(g)=>f is homoptopic to g

Let $a(t)=e^{2\pi it}$ be the generator of $\pi_1(S^1,1)$ and we define degree of $f$ this way: $[\omega^{-1}]\cdot f_*([a])\cdot [\omega]=\deg(f)[a]$ where $\omega$ is any path from $f(a)$ to $1$. ...
0
votes
1answer
47 views

Need help on finding homotopy

Define a continuous map $\ell:(I,\partial I)\to (SO(3),1)$ by $\ell(t) = \left( \begin{array}{ccc} \cos 2\pi t & -\sin 2\pi t & 0 \\ \sin 2\pi t & \cos 2\pi t & 0 \\ 0 & 0 & 1 ...
3
votes
0answers
68 views

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 ...
0
votes
2answers
63 views

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 ...
0
votes
1answer
20 views

The interior of $S^2 \setminus \{N\}$ is itself.

I am hoping to confirm that if the interior of $S^2 \setminus \{N\}$ is itself? This question is in order to satisfy Excision Theorem condition in the exercise $(D^2) \cap (S^2 \setminus \{N\}) = D^2 ...
1
vote
1answer
21 views

$(D^2) \cap (S^2 \setminus \{N\}) = D^2 \setminus \{0\}$.

I am trying to invoke Excision Theorem: For space $A, B \subset X$ whose interiors cover $X$, the inclusion $(B, A \cap B) \hookrightarrow (X, A)$ induces isomorphisms $H_n(B, A \cap B) \to H_n(X, ...
0
votes
1answer
23 views

$D^2 \cup \{pt\} = S^2$?

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
0
votes
2answers
66 views

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.
0
votes
1answer
50 views

I didn't understand this open disk question

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 ...
1
vote
2answers
47 views

Need help with computing homology group.

Let $D=$$S^2\cup$ x-axis$\cup$ y-axis be surface in $R^3$ I want to compute the homology group $H_n(D,\mathbb{Z})$ forcannot all $n\geq 0$ using Mayer-Vietoris Exact sequence. There exists many open ...
1
vote
1answer
50 views

Homology groups

I have to compute the groups $H_q(S^{3},S^1)$ (Singular Homology) I am new in the subject, i have compute some basics groups, but i dont know how to start with this one, if someone could help me, ...
2
votes
1answer
76 views

Hatcher 2.2.26 Show that if $A$ is contractible in $X$ then $H_n(X,A) \approx \tilde H_n(X) \oplus \tilde H_{n-1}(A)$

Show that if $A$ is contractible in $X$ then $H_n(X,A) \approx \tilde H_n(X) \oplus \tilde H_{n-1}(A)$ I know that $\tilde H_n(X \cup CA) \approx H_n(X \cup CA, CA) \approx H_n(X,A)$. And $(X ...
1
vote
1answer
73 views

Hatcher 2.2.31 Invoke Mayer-Vietoris to wedge sum.

Use the Mayer-Vietoris sequence to show there are isomorphisms $\tilde H_n(X \vee Y) \approx \tilde{H}_n(X) \oplus \tilde H_n(Y)$ if the basepoints of $X$ and $Y$ that are identified in $X \vee Y$ ...
0
votes
1answer
39 views

symmetry in the homotopy relation

Suppose $\alpha, \beta : I \to X$ are paths and suppose $\alpha $ is homotopic to $\beta$, $\alpha \cong \beta$. So, can find a continuous function $F(s,t) = f_t(s)$ such that $$ f_t(0) = \alpha(0) ...