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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1answer
466 views

Is path-connected and locally path-connected equivalent to simply connected?

Is path-connected and locally path-connected equivalent to simply connected? I am more confident with the part that simply connected implies path-connected and locally path-connected. And the other ...
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2answers
205 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 ...
2
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1answer
356 views

Fundamental group obtained by attaching a n-cell with n ≥ 2

I am having trouble with Hatcher's Algebraic Topology P39, Problem 18: Show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n ≥ 2$, then the ...
2
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1answer
41 views

Showing there is some point interor to the unit disk, where V=(0,0) using covering maps

Let $B$ be the closed unit disk in $\mathbb{R}^2$ and suppose that $V = (p(x, y), q(x, y))$ is a vector field (p,q are continuous functions) defined on B. The boundary of $B$ is the unit circle $S^1$ ...
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1answer
24 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 ...
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1answer
24 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, ...
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1answer
26 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
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3answers
89 views

Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style. For (connected, pointed) topological spaces with trivial ...
0
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2answers
102 views

Question on relative homology

I have this: $\ \ $ $3)$ Assume now that each critical point of $\varphi$ in $K_c$ is isolated in $X$. Let $\epsilon>0$ be such that $c-\epsilon >b$ and $c$ is the only critical value of ...
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1answer
82 views

Euler characteristic of part of the sphere

Let R be the part of the sphere in $R^3$ bounded by two smoothly closed curves that do not intersect. For instance, R is the region bounded by a great circle and a smaller circle paralle to it. How ...
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1answer
155 views

Mayer-Vietoris of pair (X,C)

I would like to know if i can use Mayer-Vietoris with this form: Let X be a topological space and A, B be two subspaces whose interiors cover X and $C\subset A\cap B$. We get the exact sequence ...
5
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1answer
159 views

Homology of the product

I have to prove that $$H_q(X\times\partial I^n,X\times\{p_0\})=H_{q-n}(X)$$ for $X$ a topological space. I tried using induction, but I didn't go too far, and think that using some exact sequence ...
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1answer
68 views

Non self-intersecting representatives in fundamental class

If $X$ is a Riemann surface with boundary $\partial X$ and $\pi_1(X,p)$ is its fundamental group, $p \in X$, then we shall call class $[\gamma] \in \pi_1(X,p)$ primitive (or generator) if it can not ...
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2answers
137 views

What is the topology of a simplicial complex?

I know what a simplicial complex is, but when reading about triangulations on surfaces I found that there must exist a homeomorphism betwen the space underlying the surface and some simplicial ...
2
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1answer
93 views

On a coalgebra structure of simplicial homology

Are there any results on the homology group of an abstract simplicial complex with coefficients in a field $k$ being a $k$-coalgebra? Are there any assumptions and restrictions on the topological or ...
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1answer
78 views

Let $X:=\mathbb R^2\setminus\{x_0,x_1\}$, where $x_0,x_1\in\mathbb R^2\setminus\{0,0\}$. Compute the fundamental Group of $\pi_1(X,(0,0))$

Let $X:=\mathbb R^2\setminus\{x_0,x_1\}$, where $x_0,x_1\in\mathbb R^2\setminus\{0,0\}$. Compute the fundamental Group of $\pi_1(X,(0,0))$ What does mean ''compute'' ? i can only draw it. or ...
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0answers
93 views

Short exact sequence of group of Hodge classes

I'm from a foreign country, I don't speak well English. Sorry. My question is : $X$ and $Y$ are subvarieties of a smooth projective variety $M$ such that $M=X \bigcup Y$. I would like to know if we ...
5
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1answer
183 views

Computing characteristic numbers of homogeneous spaces

I apologize in advance if this is a bad question. I would like to prove or refute a conjecture about the vanishing of characteristic numbers of homogeneous spaces, and to this end am looking (and ...
3
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1answer
315 views

Can the complement of a simply connected set in $\bar{\mathbb{C}}$ in an open set always be covered by a simply connected union of balls?

I believe the following to be true, but am worried my intuition does not account for fractally things: Let $K\subset\bar{\mathbb{C}}$ ($\bar{\mathbb{C}}$ being the Riemann sphere) be closed (thus ...
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1answer
88 views

Hawaiian earring is not a cofibration

Let $X$ be the union of all circles centered at $(0,\frac{1}{n})$ with radius $\frac{1}{n}$ for $n\in N$. Let $A$ be $(0,0)$. Show that $A\to X$ is not a cofibration. This appears as a non-example ...
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0answers
52 views

Brieskorn spheres that are S^3

Given pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. Can someone ...
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1answer
54 views

Definition of a one-connected manifold?

Perhaps the question is self-explanatory. The context is Kleiner's Inv. Math. paper An isoperimetric comparison theorem, where the statement of the main theorem begins with "Let $M$ be a complete ...
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2answers
445 views

Factorization of a map between CW complexes

I've been working on problem 4.1.16 of Hatcher's Algebraic Topology and am at a complete impasse. The problem is as follows: Show that a map $f:X→Y$ between connected CW complexes factors as a ...
5
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1answer
361 views

Star-shaped domain whose closure is not homeomorphic to $B^n$

A star-shaped (relative to $0$) domain $U$ is a bounded open subset of $\mathbb{R}^n$ such that for each $x \in U$, the line segment from $0$ to $x$ lies entirely in $U$. Is there a star-shaped ...
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1answer
86 views

simple question about cohomology group

Let's consider compact 4-manifold $M^{4}$ and point $P \in M$. Then (use Mayer-Vietoris) inclusion $i\colon M\setminus P \to M$ induce isomorphism $i^{*}\colon H^2(M) \to H^2(M\setminus P)$. Let's ...
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1answer
103 views

Show that $G:=\{g\ |\ g\text{ is a straight line in }\mathbb R^2 \}$ is homeomorphic to the Moebius strip

Show that $G:=\{g\ |g\text{ is a straight line in }\mathbb R^2 \}$ is homeomorphic to the Moebius strip I defined $\bar f:\mathbb R\times S^1/\tilde{}$ where $(t,\theta)\tilde{} (-t,-\theta)$ ...
5
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1answer
85 views

Why are there always pairwise intersections in a Heegaard splitting?

Let $M=A\cup B$ be a Heegaard splitting, such that $\{\alpha_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $A$, and $\{\beta_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $B$ ...
4
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1answer
126 views

Inverses in the homotopy classes of maps into $RP^{\infty}$

One can define bilinear maps $\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^{2n-1}$ by considering the elements in $\mathbb{R}^n$ as polynomials and doing multiplication. This defines an ...
2
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0answers
153 views

Set of generators of the commutator subgroup of a surface group

Good morning, I am having a hard time trying to describe the commutator subgroup of a surface group. Namely, if $S$ is a compact orientable surface and $G$ its fundamental subgroup, let's recall that ...
0
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2answers
51 views

What is the group $\Gamma$ such that $\mathbb{H}/\Gamma$ is a genus-n torus

We know that the universal cover of genus-n torus is a unit disk ($n\ge2$), which is conformal to upper half plane $\mathbb{H}$, with automorphism group $SL(2,\mathbb{R})$. Thus the genus-n torus can ...
2
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1answer
43 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
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1answer
32 views

Quick question about covering maps

Let $p:E\rightarrow B$ be a covering map and $b \in B$ so there exists a neighborhood $U$ of $b$ such that $$p^{-1}(U)=\bigcup V_\alpha \text{ (disjoint union)}$$ and each ...
3
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1answer
159 views

What is the induced map on fundamental group of the inclusion of unitary group in the orthogonal group?

What is the induced map on fundamental group of the inclusion of unitary group $U(n)$ in the orthogonal group $SO(2n)$?(Note that the unitary group $U(n)$ can only embedded in the group $SO(2n)$, not ...
2
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0answers
68 views

Triangulations of surfaces with few vertices

I'm interested in triangulations with few vertices of a given compact and oriented surface $S$. By triangulation, I do not mean a "simplicial triangulation" but a "decomposition of $S$ by ...
2
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1answer
334 views

Prove that the fundamental group of $X$ is Abelian

Let $X$ be a path-connected topological space. And there is a continuous map $F: X\times X \to X$ such that: $$F(x,x)=x \ \text{ and }F(x,y)=F(y,x).$$ Prove: The fundamental group of $X$ is Abelian. ...
5
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0answers
122 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
4
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0answers
100 views

CW structure on infinite products

There is a standard CW-topology on the finite product $X\times Y$ of CW-complexes $X$ and $Y$. Is there a standard CW-topology on an infinte prodcut $\prod_{n=1}^{\infty}X_{n}$ of CW-complexes? With ...
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0answers
57 views

Relative de Rahm cohomology computation for two disjoint circles embedded in $\mathbb{R}^2$

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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1answer
132 views

Mayer-Vietoris sequence for the figure eight

On my professor's solutions for my last algebraic topology homework, he gets the following Mayer-Vietoris sequence for the figure eight space (the wedge of two circles): $0\to H_{2}(X)\to 0\to ...
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2answers
109 views

Fundamental group of the topological space obtained by identifying the four vertices of a square

The task is: Compute the fundamental group of the topological space obtained by identifying the four vertices of a square. So we identify the vertices with the same letter. Can we say something ...
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1answer
54 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 ...
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2answers
79 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 ...
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1answer
80 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, ...
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1answer
184 views

What is an “essential loop”?

I'm a bit confused. Is an essential loop in a topological space $X$ just a loop $\alpha$, which is not-contractible (i.e. $[\alpha] \neq 0$ in the fundamental group of $X$), or is there something more ...
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2answers
237 views

Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
2
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1answer
268 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 ...
3
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1answer
31 views

When does a subspace have the same regular open algebra?

Given a topological space $X$ and a dense subspace $D$, I believe it's true that for a regular open set $U$ of $X$, $U \cap D$ is regular open in $D$. Note this induces a homomorphism between the ...
1
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1answer
79 views

Reduced homology: $\tilde H_n(X \cup CA) \approx H_n(X \cup CA, CA)$

On Hatcher 125, it says $$\tilde H_n(X \cup CA) \approx H_n(X \cup CA, CA).$$ I couldn't really see this from my understanding of reduced homology, it is just replacing $$\dots ...
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2answers
53 views

Definition of rank

In Hatcher P146, the rank of a finitely generated abelian group is defined to be the number of $\mathbb{Z}$ summands when the group is expressed as a direct sum of cyclic groups. $\mathbb{Q}$ $1$: ...
1
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1answer
92 views

An exact homology sequence associated with a principal SO(n) bundle

Suppose $P$ is a principal $SO(n)$ bundle, X is its base space. Why is there an exact sequence in homology groups $$ 0 \to H^1(X;\mathbb{Z}_2) \to H^1(P;\mathbb{Z}_2) \to H^1(SO(n);\mathbb{Z}_2)\to ...