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

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2answers
78 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
65 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
56 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 ...
2
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0answers
86 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
144 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 ...
2
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1answer
146 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
54 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
41 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 ...
2
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1answer
38 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 ...
6
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2answers
286 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 ...
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1answer
150 views

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

A star-shaped domain $U$ is a bounded open subset of $\mathbb{R}^n$ containing $0$ such that for each $x \in U$, the line segment from $0$ to $x$ lies entirely in $U$. Is there a star-shaped domain ...
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1answer
75 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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0answers
15 views

integration along fibres and kunneth

suppose $X\times I$ is a product of a smooth manifold and unit interval. There is a map $\pi_*:\Omega^k_{X\times I} \rightarrow \Omega^{k-1}_X$ called integration along fibres. Similar operation ...
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1answer
80 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
55 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
95 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 ...
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0answers
65 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 ...
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2answers
39 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
36 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
25 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
97 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 ...
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0answers
56 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
170 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
118 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
76 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
47 views

Relative de Rahm cohomology computation for two disjoint circles embedded in 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
72 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
79 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
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 ...
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2answers
55 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
56 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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0answers
65 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
113 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
90 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 ...
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1answer
22 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
53 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
50 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. $\mathbb{Q}$ $1$: ...
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1answer
55 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 ...
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0answers
87 views

Associativity in the fundamental groupoid of a space

Consider the set of all path homotopy classes of paths in $X$ with $[f]\cdot[g]=[f*g]$ defining a binary operation. We have a groupoid with the following conditions: 1) $[c_p][f]=[f]=[f][c_q]$ where ...
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0answers
21 views

Edges and genus in graphs

For a planar graph $G = (V, E)$ there is the well known bound $|E| \leq 3|V| - 6$. If instead of $S^2$ $G$ embeds in the orientable surface $S_g$ of genus $2 - 2g$ with minimal $g$, what can be said ...
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0answers
20 views

HEP of subcomplex for product topology of CW-complexes

Suppose $X$ and $Y$ are CW-complexes and $A\subset X$ and $B\subset Y$ have properties such that the product topologies of $X\times B$ and $A\times Y$ are CW-complexes, such as when $A$ and $B$ are ...
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1answer
96 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 $X \vee Y$ ...
2
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1answer
60 views

What is the center of fundamental groupoid?

$C$, $D$ are two categories. $F$, $G$ are functors between $C$ and $D$: $F, G: C\rightarrow D$. Let $Nat(F,G)$ be all the natural transformations between F and G. Like what we do for groups, define ...
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0answers
63 views

Ham Sandwich theorem used in combinatorics problem involving beads on a necklace

Ok, so according to a friend of mine you can use the ham sandwich to prove the following theorem: Suppose there is a necklace with $m$ types of beads and $2n_1,2n_2...2n_m$ beads of each colors. So ...
3
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2answers
77 views

For any element $e$ of an open set $V$ of a covering space, does there exist a sheet $S$ such that $e\in S\subseteq V$

Let $p:E\rightarrow X$ be a covering map. Let $V$ be any open subset of $E$ and $e$ be any element of $V$. I feel that the following statement must be true: There exists an evenly covered open subset ...
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1answer
97 views

Use of Low dimensional Paths vs High dimensional Cubes

The universal covering manifold has a construction as follows: Fix a base point $p$ of $M$. Two paths $c_i\colon I \rightarrow M$ ($i=0,1$) on $M$ with terminal point $p$ and the same starting point ...
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0answers
171 views

Fundamental group of a space of infinite genus and an accumulation point

The fundamental group of $\mathbb{R}^2$ with a point removed is $\mathbb{Z}$. The fundamental group of $\mathbb{R}^2$ with $n$ points removed is the free group of $n$ generators. Is the fundamental ...
11
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1answer
85 views

If the connected sum of a manifold $M$ with itself gives back $M$, does it imply $M$ is a sphere?

Let $M$ be a compact, connected, oriented $n$-dimensional manifold without boundary. Suppose that $M\#M\cong M$. Does it imply that $M \cong S^n$? Sorry if this is a naive question. This is not my ...
9
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1answer
210 views

Is the quotient map a homotopy equivalence?

It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy ...
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0answers
39 views

Show there does not exist a covering map $g \colon S^{2n} \rightarrow X,$

Show there does not exist a covering map $g \colon S^{2n} \rightarrow X,$ where $X$ is a simplicial complex with $\pi_1(X)\cong \mathbb Z/(2k + 1)\mathbb Z.$