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

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Not all finitely-presented groups are fundamental groups of closed 3-manifolds

It is a well-known result that, for any finitely-presented group $G$ and any integer $n \geq 4$, there exists a closed $n$-manifold whose fundamental group is isomorphic to $G$ (a sketch of proof can ...
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15 views

Is a knot group torsion-free?

It was mentionned in a previous discussion that it is unknown whether or not the fundamental group of an open connected subset $X \subset \mathbb{R}^3$ is torsion-free. But is it possible to answer ...
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0answers
9 views

The identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$

Show that the identity map $\Delta^n \rightarrow \Delta^n$ is a basis for $H_n(\Delta^n, \delta\Delta^n; R)$. Here $\Delta^n$ is the n-simplex, and I know $\delta \Delta^n$ denotes its ...
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1answer
21 views

Is composition of covering maps covering map?

In Munkres book, composition of covering maps is covering map when $r^{-1}(z)$ is finite for each $z$ in $Z$ where $q : X\to Y$ , $r:Y\to Z$ are the covering maps. I tried hard to find an example that ...
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1answer
19 views

How to Prove that these Spaces are not Homotopically Equivalent

Let $X=\{0\}\cup\{1/n:n\in\mathbb{N}\}$ and $Y$ be any countable discrete space. Show that $X$ and $Y$ are not homotopically equivalent. The hint says: Every continuous map from $X$ to $Y$ takes all ...
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22 views

The generator of compact cohomology of punctured plane

Can anyone give detailed derivation of the generator of compact cohomology of $H^1_c (R^2-\{0\}$). (It is homotopic to a circle so it is isomorphic to $R$ but I want computation of its generator, ...
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1answer
24 views

What does “minus the zero section” mean?

I see a vector bundle obtained from another one by the means of "minus the zero section" in some literature. The concept zero section of a vector bundle is found in Section Sections and locally free ...
2
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1answer
31 views

the 0-th homology of a simplical complex

Let $K$ be an (abstract) simplicial complex. The claim is: $H_0(K;\mathbb{Z})$ is always nonzero. Is this possible to prove it without any "special techniques to computing homology-groups"? ...
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0answers
17 views

Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
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1answer
22 views

What is the $\pi_1$-action on the hom-sheaf between two finite etale covers?

Say you have two finite etale covers $X\rightarrow S$, $Y\rightarrow S$. The hom sheaf $\mathcal{H}om_S(X,Y)$ on the etale site $\text{Sch}/S$ is finite locally constant, hence representable by some ...
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2answers
31 views

How to compute the fundamental group of this space?

I know that without the closed disk, a sphere with the a diameter deformation retracts onto the wedge sum of a circle and a sphere. But I can't figure out how to deform the disk to a suitable ...
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1answer
30 views

Any invertible linear map is homotopic to a composition of reflections

I am trying to solve a problem in Hatcher. I reduced my problem to showing that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear map, then $f$ is homotopic to a composition of reflections, ...
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1answer
24 views

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$.

Let $S_ 0$ be the space with $2$ points and the discrete topology. Find [$S_ 0$ , $X$] for an arbitrary space $X$. $[X,Y]=\{f:X\to Y,f$ continuous $\}/\sim$ where $\sim$ is the homotopic equivalence. ...
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1answer
21 views

Prove that $C_1$ and $C_2$ are homotopic fixing endpoints.

Let $C_1$ and $C_2$ be two great circles in $S^2$, intersecting at the points $p,q$. If we consider $C_1$ and $C_2$ as curves starting and ending at $p$. Prove that $C_1$ and $C_2$ are homotopic ...
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1answer
40 views

Relation about Disk and Sphere

Definition of sphere and disk are following \begin{align} S^n =\{ (x_1 , \cdots x_{n+1}) \in \mathbb{R}^{n+1} | \sum x_i^2 =1 \} \end{align} \begin{align} D^n =\{ (x_1 , \cdots x_{n}) \in ...
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1answer
52 views

What categorical limits and colimits does $\pi_1$ preserve?

$\pi_1$ is a functor from the category of pointed topological spaces to the category of groups. It's clear that this functor preserves products, since $\pi_1(X\times Y)=\pi_1(X)\times\pi_1(Y)$, and a ...
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1answer
40 views

Dual notion of the subspace topology

Let $X$ and $Y$ be sets with $\iota:X\to Y$ an injection. If $Y$ is a topological space, we define the subspace topology on $X$ as the initial topology induced by this diagram. Analogously, if $X$ ...
2
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1answer
26 views

Homotopy groups of infinite Grassmannians

Let $G_k$ be the infinite rank $k$ Grassmannian. For $n>k$, is $\pi_n(G_k)$ trivial? Phrased differently, is every rank $k$ vector bundle on an $n$-sphere, for $n>k$, trivial? (This is motivated ...
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1answer
17 views

Proof explanation Mayer-Vietoris and the Punctured Euclidean Space.

I am only releasing part of the proof. Doesn't this prove that "otherwise" case is completely wrong? I am assuming "otherwise" refers to $n < 2$? Because he just showed that $H^0(\Bbb ...
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0answers
24 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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2answers
54 views

Why exactly is this injective? Algebraic Topology.

Let $(A^*, d^i)$ be a chain complex of finite dimensional vector spaces, i.e, $$0 \to A^0 \to A^1 \to \dots \to A^n \to 0.$$ Show the sequence $$0 \to H^i(A^*) \to A^i/Im(d^{i-1}) \to Im(d^i) ...
6
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1answer
49 views

Serre fibrations vs. Hurewicz fibrations

What is an example of a Serre fibration that is not a Hurewicz fibration?
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37 views

The Complex projective space is homeomorphic to the n-sphere

Ok I have been asked to give as detailed a proof as I can for the following question. Prove that $ \mathbb C\mathbb P^n $ is homeomorphic to $ S^{2n+1} /\sim. $ where for $ z,w \in S^{2n+1} \subset ...
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1answer
58 views
+100

Problem with understanding proof of Van Kampen's theorem

I'm currently reading J.P May's book, "A Concise Course in Algebraic Topology". I don't understand his proof of the fundamental groupoid version of Van Kampen's theorem, particularly the part where ...
4
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0answers
29 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
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2answers
51 views

Definition question in algebraic topology.

Definition: The $p$th (de Rham) cohomology group is the quotient vector space $$H^p(U) = \frac{Ker(d:\Omega^{p}(U)\to \Omega^{p+1}(U)}{Im(d:\Omega^{p-1}(U)\to \Omega^{p}(U))}$$ where $U \in ...
2
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1answer
31 views

How to show that a homomorphism between fundamental groups is an inner automorphism?

I want to show that if $\sigma$ is a loop based at a point $p \in X$, where $X$ is a topological space, then the homomorphism $\Phi _\sigma: \Pi_1(X,p) \rightarrow \Pi_1(X,p)$, is an inner ...
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1answer
73 views

Fibrations lift p-connectedness

Let $p: X \to B$ be a Serre fibration, and suppose that $B^p \subset B$ is a subspace of $B$ such that $(B,B^p)$ is $p$-connected, i.e. $\pi_n(B,B^p)=0 \ \forall n \leq p$ or, equivalently, ...
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2answers
61 views

if $\pi_1(G)$ is trivial, how to prove $ \pi_1(G/H)=\pi_0(H)/\pi_0(G) $?

Let $G$ be a topological group, $H$ be its normal subgroup of $G$, and $G/H$ be the quotient space induced by the natural map.(We know that $G/H$ is again a topological group) If $\pi_1(G)$ is ...
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0answers
24 views

Free action of symmetric group

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$? Is there a compact manifold which can be act freely by all symmetric ...
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0answers
33 views

Explicitly building Top. space with trivial homology group and non trivial fundamental group, w/o CW-complexes

I know this is a pretty famous question here, but I was asked to show explicitly such space, during a bachelor lecture, without using any CW-complex result. I started working using some covering ...
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0answers
13 views

Homotopy classes of self-maps on L(2,1)#L(3,1)

How to classify continuous self maps on $L(2,1)\# L(3,1)$ up to homotopy? Here $L(2,1)$ and $L(3,1)$ are lens spaces. The reason for considering this manifold is that its fundamental group is ...
4
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1answer
45 views

Calculating $\pi_2$ of a certain free loop space

For a topological space $X$, define $LX$ to be the set of continuous maps $S^1 \rightarrow X$ with the compact-open topology. Henceforth let $X = \Bbb{CP}^\infty \times \Bbb{RP}^\infty$, with ...
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2answers
97 views

When is the free loop space simply connected?

I am not sure if there is an obvious answer to this, but this has been bothering me. Let $X$ be a topological space. When is the free loop space, $LX$, simply connected? Correct me if I'm wrong, but ...
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1answer
39 views

Is there no continuous $f:\mathbb{R}^{2^{*}}\rightarrow S^1$?

This is a question from last year's exam: Prove that there is no continuous $f:\mathbb{R}^2 -\{0\}\rightarrow S^1$ such that $f(x)=x$ on $S^1$. Well this question is equivalent to show that ...
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2answers
25 views

equivalent characterisation of simply connect spaces

I want to prove the following: Let $X$ be path connected space, $S^{1}$ the $1$-sphere and $D^{2}$ the unit circle. Following are equivalent: i)X is simply connected. ii) If $f:S^{1} \to X$ ...
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2answers
28 views

Pathwise, simple connectedness of real Grassmannian $G(2, 4)$

Let $G(2, 4)$ denote the space of two dimensional planes in $\mathbf R^4$. I have found that the integral homology is the following: $H_0 = \mathbf Z, H_1 = \mathbf Z / 2 \mathbf Z, H_2 = 0, H_3 = ...
11
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1answer
150 views
+50

Functoriality of fundamental group

I'm trying to prove the following statement: If $\mathcal{C}_1$ is the category of path connected topological spaces and $\mathcal{C}_2$ is the category of groups, then the mapping $\mathcal{C}_1 \ni ...
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2answers
24 views

is there a notation to designate the induced homomorphism including base point?

Let $X,Y$ be a topological spaces. Let $f:X\rightarrow Y$ be a continuous function. Fix $x_0\in X$. Define $f_*([r])=[f\circ r]$ for every loop $r$ at $x_0$. Then, $f_*:\pi_1(X,x_0) \rightarrow ...
11
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2answers
107 views

Is the plane minus the integer lattice homeomorphic to the plane minus the integers?

The question, more rigorously posed, is: Is $\Bbb R^2-\Bbb Z^2$ homeomorphic to $\Bbb R^2-\Bbb Z\times\{0\}$? This question has been bugging me in the back of my head for a while now. Sometimes, ...
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0answers
28 views

How do I prove that the union of two simply connected open sets whose intersection is path connected is simply connected?

I'm trying to understand Ronnie Brown's answer here: union of two simply connected open , with open and non empty intersection in $R^2$ Let $X$ be a topological space and $U,V$ be simply connected ...
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1answer
30 views

What is the definition of contractible space? (It is not a duplicate)

I'm studying two algebraic topology texts (namely Munkres and Theodore) Here are definitions given in those texts Munkres Let $X$ be a topological space. If the identity map on $X$ is ...
2
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0answers
19 views

finite simplicial complex compact

Let $K=(V,\Sigma)$ be a finite simplicial complex. I want to show that $|K|$ is compact. I know that $K$ is a sub-simplicial complex of $\Delta^V$ with $|\Delta^V|$ compact. So I think I should show ...
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0answers
20 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
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1answer
28 views

The space of $S^1/S^1$, the space of a single point, and their first homotopy group

I read from the book Soft matter physics by Kleman that the space $R$ of a point is $0$ and its first homotopy group $\pi_1(0)=0$. This causes some confusion to my understanding. Why the space of a ...
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0answers
32 views

abelianized fundamental groups.

I am trying to show that there is a canonical ismorphism between the abelianized fundamental groups, $\pi_1(X,p)_{ab}$ and $\pi_1(X,q)_{ab}$ of the path-connected space $X$. I know since $X$ is path ...
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0answers
53 views

Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the ...
1
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1answer
18 views

Fundamental group of Antoine's necklace

Let $A \subset \mathbb{R}^3$ denote Antoine's necklace. It is well-known that $A$ is a Cantor space and that $\mathbb{R}^3 \backslash A$ is not simply connected. Futhermore, $\pi_1(\mathbb{R}^3 ...
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0answers
14 views

whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
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0answers
21 views

classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...