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Questions tagged [algebraic-topology]

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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Fibers Under a Covering Map are Discrete Subspaces of the Domain

In Munkres' topology book, the following claim is made: If $p : E \to B$ is a covering map, then for every $b \in B$, $p^{-1}(b)$ is a discrete subspace of $E$. Here's my attempt at a proof: ...
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Explicit determination of the open book in $S^3$

I'm familiar with the fundamental concepts of algebraic and differential topology. How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: \mathbb{C}^...
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Can we use Eckmann–Hilton duality to prove dual theorems?

I'm reading Tammo tom Dieck's algebraic topology text. In the text the Eckmann–Hilton duality is mentioned several times. My current understanding is that it captures the "equivalence" between a ...
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1answer
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Why is the classification of topological spaces up to homeomorphism impossible or undesirable?

In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or ...
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1answer
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Relative homology and limit

Let $X$ be a smooth manifold and $O\subset X$ be an closed set containing a non-trivial neighbourhood of $x\in X$. The reason to ask the question is to clarify the relationship between limit and ...
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1answer
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Reversing a Path Formed by Concatenation

Let $X$ be some topological space, let $a,b,c$ be points in $X$, let $f$ be the path from $a$ to $b$, and let $g$ be the path from $b$ to $c$. Define $h = f * g$. Then $\overline{h} = \overline{g} * \...
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1answer
661 views

Does there exist a continuous 2-to-1 function from the sphere to itself?

I am interested in the following question: Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$? I suspect the answer is no, but I don't know ...
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Find the fundamental group of the following spaces

Find the fundamental groups of the following spaces. In each case they can be built up from cyclic groups by free products and direct products. (a) The space obtained from two copies of the torus $S^...
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1answer
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Local Path Connectedness - Collection Of All Path Connected Open Sets Is A Basis

A space $X$ is locally path connected if $X$ has a basis of path connected open sets. A follow-up to this question: Hatcher Universal Covering Space Construction - Basis From this definition, $X$ ...
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23 views

Existence of Thom Class

In page 133, Theorem 8.5.5. (The Thom isomoprhism theorem) Let $\pi:V \rightarrow X$ be a complex vector bundle of rank $n$, over al locally comapct space $X$. Let $$ 0 \rightarrow \pi^* \...
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Relative homology: the splitting $C_*(X,A) \rightarrow C_*(X)$

Let $C_n(X)$ be the free Abelian group generated by all the singular $n$-simplices of the topological space $X$. I'm reading Bredon's Topology and Geometry and there's a statement which says: The ...
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Defining Bott Class by relative $K$-theory

I am really confused with this construction of Bott Class in Page 127, Example 8.4.12 If $V$ is a complex vector space of dimension $n$, we form the complex $$ 0 \rightarrow \wedge^0 V \...
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2answers
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Hatcher Universal Covering Space Construction - Basis

Below is an excerpt from Hatcher's Algebraic Topology. He is constructing a universal cover for a path-connected, locally path-connected, and semilocally simply-connected space $X$: I don't ...
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Free topological $G$-space and G-CW-Complex

Let $G$ be a finite group, and $X$ a free topological $G$-space which admits a CW-structure. Is there a CW-structure on $X$ that compatible with its $G$-action, i.e, a cell structure that turns $X$ to ...
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1answer
26 views

Topology of decomposition of a space

In pg 121 of this notes, the author outlines a construction of gluing bundles. The scenario begins with Let $X= X_0 \cup X_1$ be union of two comapct spaces. $A = X_0 \cap X_1$ so that $X = X_0 \...
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1answer
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Hatcher Lemma 1A.3 - Covering Spaces Of Graphs Are Graphs

Below is an excerpt from Hatcher's Algebraic Topology: There are a few things I don't understand about the highlighted red part. What exactly is a "basic open set"? The highlighted sentence seems to ...
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A difficulty in understanding the proof of boundary theorem in G&P.

The theorem and its proof is given below: But I could not understand the last line in the proof in particular: Why $F^{-1}(Z)$ is a compact one dimensional manifold with boundary? And why this leads ...
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Computing homology groups

Given two copies of $D^2 \times S^1$ (Full torus) glued along the boundaries by a map from a Torus to itself defined by the action of $M \in SL(2, {ℤ})$; i.e., the map is $(x,y) \mapsto (x^ay^b, ...
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1answer
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Cohomology of Projective Space $\mathbb{PR}^n$ with Coefficients in $\mathbb{Z}/2$

We know the cohomology ring with coefficients in $\mathbb{Z}/2$ of projective real space $\mathbb{PR}^n$ is $$H^*(\mathbb{PR}^n, \mathbb{Z}/2) = \mathbb{Z}/2[X]/(X^{n+1})$$ with graduated generator ...
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pell's equation converge

Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.
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(Co)homology of $S^2×S^2/ℤ_2$

Cohomology of $S^2\times S^2/\mathbb{Z}_2$ I was looking at this question, the accepted answer uses the homology of the space to find the cohomology. I was wondering how one could compute the ...
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0answers
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How to compute the local degree for a specific function

Using the definition of the local degree from Hatcher pg. 136, how can we explicitly calculate the local degree of a map? For example, to calculate the local degree of a map $f:R^2 \rightarrow R^2$ ...
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Inverse problem : Finding a vector bundle (resp. with connection), given its characteristic class (resp. differential character)

Given a rank $n$vector bundle $\alpha :E \to M$, and an element $u \in H^k(BG, \mathbb{Z})$, $G=GL(n,\mathbb{R})$we can define its characteristic class $u(\alpha) \in H^k(M, \mathbb{Z})$ as $f_\alpha^*...
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Calculate deRham Cohomology un R^3

I want to calculate the Cohomology groups of $B_r-B_s$ $(r>s)$ where $B_r$, $B_s$ are solid balls on $\mathbb{R}^3$ and the boundary of $B_s$ is empty, I tried use Mayer Vietoris and the $U$,$V$ ...
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1answer
29 views

Barycentric subdivision preserves geometric realization

I have the following definitions: Definition 1: A simplicial complex $K$ is a family of finite nonempty subsets of a set $V_k$ (the elements of $V_k$ are called vertices) such that: 1) if $v\in V_k$,...
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0answers
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Loring Tu.Second edition Excersice 28.6

I need help with this problem: Compute the cohomology of $\Sigma_2 \setminus \{p\}$ where $\Sigma_2$ is a surface of genus 2 and $p$ is one point. I understand the idea in this chapter but I will ...
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advice for beginner in representation-theory

Thank you for you reading.I am in my second year of my undergraduate and I plan to study representation-theory in the future. I had courses in linear algebra, abstract algebra(Hungerford chapter I-V ...
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1answer
42 views

An alternative, more formal proof of a path lifting criterion in tom Dieck's Algebraic Topology

This is a theorem from Tammo tom Dieck's Algebraic Topology: While it has a direct proof, the author gives a more formal proof in the problems: By pullback I suppose he means a diagram $\require{...
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27 views

Notation on fibre bundles

I came up this morning with the following question and after looking for it for a while on the internet i found this old question on math.stackexchange with no answers. Could anyone please give some ...
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1answer
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de Rham cohomology of doubly punctured torus

Let $T^2=S^1\times S^1$. I'd like to know all de Rham cohomology groups of $M=T^2-\{a,b\}$ but I couldn't find a result. So I want to compute it and I'm thinking of using Mayer Vietoris sequence. I ...
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37 views

the index of a closed curve is continuous

Suppose $\gamma:[a,b]\rightarrow\mathbb{C}$ is a closed curve in the complex plane.We know that if $f:\widehat{\gamma}\rightarrow\mathbb{C} $ continuous such that $f(\gamma(t))\not=0 $ $\forall t\in[a,...
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The fundamental group of a topological space is isomorphic with its connected component fundamental group

can you help me with this problem of fundamental groups? suppose that $X$ is a topological space, let's fix a point on $X$ like $p\in{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ ...
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1answer
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Kähler manifolds are formal

I want to understand why Kähler manifolds are formal. This was first proved by Deligne, Griffiths, Morgan, Sullivan Let $\mathcal M$ be a minimal differential algebra and $H^*(\mathcal M)$ the ...
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1answer
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coverage mapping at covering spaces

the professor at university asked us: is a coverage mapping like P from X to Y a closed mapping or not. Also; is p an open mapping? i could prove that P is an open mapping but for proving that P is ...
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Homeomorphism in compact two dimensional manifold, periodic points, and Euler Characteristic.

I want to prove that if a homeomorphism (a continuous bijection with continuous inverse) in a two dimensional manifold doesn't have a periodic point, then the Euler Characteristc of the manifold is ...
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1answer
86 views

Compute $\pi_2(\mathbb{S}^2,X)$ where $X$ is the figure 8

We have the following short exact sequence from the long exact sequence for a pair $$0\to\pi_2(\mathbb{S^2})=\mathbb{Z}\to\pi_2(\mathbb{S}^2,X)\to\pi_1(X)=F_2\to0.$$ I wanted to construct a section (I ...
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1answer
101 views

Existence of transverse homotopy between knots in a 3-manifold

I have a 3-manifold $\Sigma$ and two homotopic embedded knots $K_{0}(t): S^{1} \to \Sigma$ and $K_{1}(t): S^{1} \to \Sigma$. I wish to refine the homotopy between them to a "transverse homotopy" i.e, ...
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1answer
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(Co-)fibrations in Top and CGWH

Suppose that you have a map $i: A\rightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a ...
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1answer
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Boundary Map in Mayer Vietoris and Homology of Knot Complement

Let $K$ be a knot in $S^3$, and N(K) be its tubular neighborhood. I want to compute the homology of $S^3-N(K)$ using Mayer-Vietoris. Let $A$ be $N(K)$ minus a small neighborhood, and $B$ be the ...
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4answers
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Homological algebra using nonabelian groups

Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice ...
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1answer
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Having problem with tom Dieck's algebraic topology text

(An online PDF of the text Algebraic Topology by Tammo tom Dieck can be found here.) This question is really soft. I'm having problem reading this text. Let me elaborate. I found this book too ...
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What is the theorem being mentioned here?

In this video, at just after the 5 minute mark, the speaker says: "...the colimit of the diagram has the homotopy type of the homotopy colimit. Why? Because all of the maps included in the diagram ...
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Non-existence of a generic solution to system of nonlinear equations

I have the following system of nonlinear equations: $f_1(x_1,..,x_m,y) =0$ $...$ $f_n(x_1,..,x_m,y) =0$ where $f_i(\cdot)$ is a nonlinear, (infinitely) differentiable equation (but not polynomial),...
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Morse Homology of $\mathbb{R}\mathbb{P}^2$

I'm trying to compute the homology of $\mathbb{R}\mathbb{P}^2$ using the following Morse function. First, consider $f:S^2 \to \mathbb{R}$ taking $(x,y,z) \mapsto y^2 + 2z^2$. This can be shown to be ...
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1answer
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Showing $\langle a,b\mid abab^{-1}\rangle$ and $ \langle c,d \mid c^2d^2\rangle$ are isomorphic.

I computed the fundamental group of the Klein bottle in two different ways and obtained two seemingly different answers: $$ \langle a,b \mid abab^{-1}\rangle $$ and $$ \langle c,d \mid c^2d^2\rangle. $...
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1answer
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Proving that $X = A \cup B$ is a disconnection then $H_n(X) \cong H_n(A) \oplus H_n(B)$ using excision

If $X = A \cup B$ is a disconnection then $H_n(X) \cong H_n(A) \oplus H_n(B)$ If I use the Mayer-Vietoris sequence I can prove this easily, however the book I'm reading hinted at using excision to ...
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Proof of Borsuk-Ulam Theorem using singular Homology

I have to prove the Borsuk-Ulam theorem following some specific steps. The theorem says that: For every continuous function $f: S^n \rightarrow \mathbb{R}^n$ there exists $x \in S^n$ with $f(-x)=f(...
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1answer
51 views

Formality of commutative differential graded algebras

I want to understand the definition of a commutative differential graded algebra (CDGA) to be formal. Actually I encountered two definitions, but I have trouble with both. From Wikipedia: A ...
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0answers
16 views

Necessary and sufficient conditions for a set to lie in a hemisphere.

Assume $A\subset S^2$ a closed subset. I am interested in necessary and sufficient conditions for $A$ to lie in a (open or closed hemisphere). For example, it is necessary for $A$ to not contain ...
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Is a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle over a 3-manifold $M$ equivalent to an element in $H^1(M,\mathbb{Z}_2)\times H^2(M,\mathbb{Z}_4)$?

Given a 3-manifold $M$ and a principal $\mathbb{Z}_2\ltimes PSU(4)$-bundle $P$ over $M$ whose isomorphism class is represented by the homotopy class of a map $f:M\to B(\mathbb{Z}_2\ltimes PSU(4))$ ...