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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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Sheafs with $F(X)=\emptyset$

In the book "A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology" by Jean Gallier, the author states on page 214 that if F is a sheaf on a topological space X and for an open set U, $F(...
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Prove that the intersection of the free boundary minimal surface with sphere is a circle.

imagine there is a free boundary minimal surface $\Omega \to R^{3}$ in unit ball $B^{3}$ how to prove that the intersection curve of this minimal surface with unit sphere $S^{2}=\partial B^{3}$ which ...
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Proving different projective planes homeomorphic?

I am having major trouble showing that the version of the projective plane here (with a Mobius strip) is homeomorphic to the projective plane that is defined as the quotient of the sphere $S^2$ via ...
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Gluing a finite number of triangles together to make a genus 2 torus (an octagon surface)? [duplicate]

How would one go about showing or describing this? A finite number of triangles glued together to make an octagon? Thank you for any help/proofs.
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Can every function which is not null-homotopic be detected by some cohomology theory?

Let $f\colon X\to Y$ be a continuous function which is not null-homotopic. This doesn't necessarily mean the induced map on singular cohomology is non-zero: for example if $c\colon S^{2n+1}\to BO(2n+1)...
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Odd maps from $S^1$ to $S^1$ viewed from the quotient $\mathbb{R}/\mathbb{Z}$ perspective

Reading a proof about degree oddity of odd continuous maps from $S^1$ to $S^1$, I have the following: Let $f : S^1 → S^1$ be an odd map : $\forall z ∈ S^1 ⊂ \mathbb{C}, f (−z) = −f (z)$ $\...
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Proving that a continuous map of topological spaces induces a homomorphism of fundamental groups

I just need a check on this. I feel that the proof is weak, but I want to make sure. Let $\phi: X \rightarrow Y$ be a continuous map of topological spaces. Define $\phi_\ast: \pi_1 (X, x) \...
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1answer
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The pre-image of a set boundary is included in the boundary of the pre-image of the set: the $R^m$ case

Premise: I am not a mathematician, nor an expert in topology. So sorry for the dumb question, if dumb it is. Consider a smooth function $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$, and a closed subset $...
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Question about excision

Let $(X,A)$ be a compact relative $n$-manifold such that $X-A$ is orientable. Let $V$ be an open neighborhood of $A$ with $V$ contained in the interior of $N$ ($N$ is a closed neighborhoods of $A$). ...
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Functions from an $n$-dimensional hypercube to $\mathbb{R}^m$ when $n >m$.

Let $n$ and $m$ be integers such that $n > m$. Suppose there exists a $n$-dimensional hypercube in $\mathbb{R}^n$. Let the hypercube be divided into $2^n$ regions ($n$-dimensional volumes) by ...
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What is the fundamental group of 4 lines with endpoints identified?

Given $4$ $1-$simplex, put an orientation on each of them, identified all of their starting and end points together. How to calculate the fundamental group of this space? I tried Van-kampen by taking ...
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1answer
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Universal property of the homotopy limit/colimit.

I have been trying to find a reference for what I have heard is a universal property which defines homotopy limits and colimits. In the category Top, colimits can be defined using the following ...
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1answer
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Hatcher Lemma 1.19

Let $f$ be some loop about $x_0$. From what I understand, we want to show that $\varphi_{0*}([f]) = \beta_h\varphi_{1*}([f])$ or $[\varphi_{0} f] = [h \ast (\varphi_{1}f) \ast \overline{h}]$ From what ...
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Finite Simplicial Complexes: Show that open sets have open connected subsets containing any point

The question: I am required to show the following: Let $K$ be a finite simplicial complex, let $x \in |K|$ and suppose that $U$ is an open set containing $x$. Then there is an open connected set $V$ ...
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Why is homology of pointed spaces the same as reduced homology of the space.

Let X be a non-empty space and $x_0\in X$, then we have that $H_k(X,x_0)\cong\tilde{H}_k(X)$. (This Statement can be found in 'Algebraic Topology' from Hatcher or slightly different in 'Topology and ...
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1answer
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Does the triviality of the orthonormal frame bundle imply the triviality of the spin bundle?

Let $M$ be a space and time orientable spin semi-Riemannian manifold of signature $(p,q)$, ${\rm Fr}(M)$ be its bundle of space and time oriented pseudo-orthonormal frames, $\Lambda : P\rightarrow {\...
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1answer
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First fundamental group and connectedness of $X:=\mathbb{R^4}\setminus \{\pi_1 \cup \pi_2 \}$

On $\mathbb{R^4}$ consider $\pi_1 := \{x_1=x_2=0\}$ and $\pi_2 :=\{x_3=x_4=0\}$. Let $X:=\mathbb{R^4}\setminus \{\pi_1 \cup \pi_2 \}$ . Show that $X$ is arc-connected and find $\pi_1 \left(X\right)$...
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1answer
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Topology on the space of functions

Exercise Question ii) I don’t manage to prove that the intersection of open sets is still open. Let’s assume that $U_1,U_2,...,U_m$ are in T. Let $f_0$ be in the intersection of $U_1,U_2, ...,U_m$. ...
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1answer
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Embedding Dimension of a graph

Given a graph $G = (V, E)$, let the embedding dimension of a graph be the least number $n$ such that there exists a partitioning of $\mathbb{R}^n$ with the contact graph is $G$. Is there any way to ...
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Canonical section of a Hirzebruch surface

What is the definition of the canonical section of the Hirzebruch surface $\mathbb{F}_2=\mathbb{P}(\mathcal{O}(-2)\oplus \mathcal{O})$?
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fundamental group of $ \mathbb{S}^3/\mathbb{Z}/3\mathbb{Z}$

Consider $\mathbb{S}^3 \subset \mathbb{C}^2$ as the set of $(u, v)\in \mathbb{C}^2$ verifying $|u|^2 + |v|^2 = 1$ Consider the action of $\mathbb{Z}/3\mathbb{Z}$ on the sphere $\mathbb{S}^3$ given ...
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2answers
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Is a closed embedding of CW-complexes a cofibration?

It is a standard fact that the inclusion of a sub-CW-complex into a CW-complex is a cofibration, it follows from the fact that the inclusions $S^k\to D^{k+1}$ are, and that they are preserved by ...
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1answer
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How does Hatcher conclude that the set of points where $\tilde f_1$ and $\tilde f_2$ agree is both open and closed?

The last line of the proof he notes that the set of points where $\tilde f_1$ and $\tilde f_2$ agree is both open and closed. How does this follow from the argument?
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1answer
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Operation with graphs

I´m trying to apply the Euler theorem (V+F=E+2 on plane graphs) for a concrete graph (the following one). Thing is that I can use to operations: 1) Delete a node of grade 2 (that is, 2 edges on it) 2)...
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1answer
48 views

Loops as maps from $S^1\to X$, Hatcher 1.15

I am working on Hatcher's problem 1.1.5. Show that for a space $X$, the following three conditions are equivalent. $\textit{a)}$ Every map $S^1\to X$ is homotopic to a constant map. $\textit{b})$ ...
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1answer
46 views

Show that a closed path is homotopic to a constant path

Let $\delta$ in $(\mathbb{R}^2-\alpha)$ where $\alpha$ is a path joining $0$ to $\infty$ which is injective, i.e., it is a simple path, it does not intersect itself. Show that $\delta$ is homotopic to ...
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1answer
42 views

Spaces of submanifolds

Let $M$ and $N$ be smooth manifolds with $\dim M<\dim N$. The spaces $\mathrm{Emb}(M,N)\subset\mathrm{Imm}(M,N)$ of smooth embeddings and immersions $f:M\to N$, respectively, are infinite ...
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What needs to be done to prove that a vector bundle is trivial?

What needs to be done to prove that a vector bundle is trivial? equivalently, This can be thought of as proving that the vector bundle satisfies the criteria of being trivial, then what is this ...
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1answer
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If $f_0=\omega_m$ and $f_1=\omega_n$, why is it not automatic that $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$?

In the second paragraph of Hatcher's proof, fourth sentence, it says The uniqueness part of (a) implies $\tilde f_0 = \tilde \omega_m$ and $\tilde f_1 = \tilde \omega_n$. How does the uniqueness part ...
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1answer
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Commutativity of fibrations up to homotopy implies commutativity

It is well-known that Suppose that $i:A\to X$ and $i': A\to X'$ are cofibrations and $g:X\to X'$ is a homotopy equivalence so that $g\circ i\simeq i'$ (commutes up to homotopy). Then $g$ is ...
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1answer
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Is this inclusion a cofibration?

Let $(X,A)$ be a relative CW-complex. Consider the inclusion $$i:\ \ X\times \{0,1\} \cup A\times I \rightarrow X\times I.$$ I was wondering if this is a cofibration. I guess it is, for there is a ...
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1answer
47 views

Why does any connected closed $m$-manifold that can be embedded in $E^{m+1}$ bounds a compact connected $(m+1)$-manifold?

I am reading Sheila Carter and S.A. Robertson's paper Relations Between a Manifold and its Focal Set. In this paper, they use the following facts: Any closed $m$-manifold $M$ that can be embedded ...
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An open cover of $\mathbb{R}^n$ and $\mathbb{C}^n$

Consider the following subset of $\mathbb{R}^n$: \begin{eqnarray}V_i:=\{(p_1, \cdots, p_n)\in\mathbb{R}^n|x^n-p_1x^{n-1}+p_2x^{n-2}-\cdots+(-1)^np_n=0\text{ has at least one root with multiplicity at ...
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1answer
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Is the complex cobordism spectrum, $MU$, a finite spectrum?

Is the complex cobordism spectrum, $MU$, a finite spectrum? If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite? What about the connective $K$-theory $...
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Examples of weak monoidal Quillen equivalences

Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen ...
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1answer
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Homology of Klein bottle with Mayer-Vietoris

I'm practicing with using the Mayer-Vietoris sequence, and found this computation. I thought it would be a good exercise to try cutting the Klein bottle into two cylinders, instead of into two mobius ...
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1answer
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Finding an open simply connected subset in a punctured open simply connected set

Let $X$ be an open , simply connected (path connected with trivial first homotopy group) subset of $\mathbb R^2$. Let $0\in X$. Is it true that for every $p,q\in X\setminus \{0\}$, there is an open ,...
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1answer
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When does $(F,f): (X,A) \rightarrow (Y,B)$ induce isomorphisms on homology?

Consider a map of topological pairs $(F,f): (X,A) \rightarrow (Y,B)$ consisting of a map $F: X \rightarrow Y$ and a map $f: A \rightarrow B$ such that $f$ is the restriction of $F$ from $A$ to $B$. We ...
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1answer
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$H_k (S^m, S^m \setminus x)\cong H_k (S^m, pt)$?

Why does for all $k\in Z$ $H_k (S^m, S^m \setminus x) \cong H_k (S^m, pt)$ (for $x\neq pt)$ hold? The inclusion $(S^m, pt) \rightarrow (S^m, S^m \setminus x)$ might induce an isomorphism on homology, ...
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1answer
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$H_k (R^m, R^m \setminus 0) \cong H_k (S^m, S^m \setminus x)$?

Why does (for singular homology) for all $k\in Z$ $H_k (R^m, R^m \setminus 0) \cong H_k (S^m, S^m \setminus x)$ for some point $x \in S^m$ hold? I thought that this could follow from the fact that $R^...
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1answer
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Construction of Moore Space

While reading the construction of Moore space from Hatcher's Algebraic topology on page 143 , I faced the following problem:--- Let $G$ be an abelian group and $0\rightarrow K\rightarrow F\rightarrow ...
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1answer
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Let $X$ be a connected CW complex and $G$ a group such that every $\pi_1(X)\to G$ is trivial. Show that every $X\to K(G, 1)$ is nullhomotopic.

Question 2 in Chapter 1.B in Hatcher's Algebraic Topology: Let $X$ be a connected CW complex and $G$ a group such that every homomorphism $\pi_1(X)\to G$ is trivial. Show that every map $X\to K(G, ...
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1answer
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How is this circle in $S^1 \times D^2$ null-homotopic?

This picture is from Hatcher's AT: I have been told that this circle is null-homotopic, but I can't see why. I know $S^1 \times D^2$ is a solid torus, but $A$ is linked with itself. How are we to ...
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1answer
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Why is the fundamental group of the Hopf link abelian but a two component unlink isn't

Title says it all basically. I'm trying to understand why the fundamental group of the hopf link (or really, the compliment of the Hopf link) is abelian. I mean, in a certain way I understand it, but ...
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Is there a Mayer-Vietoris sequence for an (uncountably) infinite collection of sets?

In Allen Hatcher's Algebraic Topology, Van Kampen's theorem is stated for a (possibly uncountably infinite) collection of path-connected open sets $A_\alpha$ whose union is some topological space $X$. ...
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1answer
80 views

Classification of contractible 4-manifolds

Is there a general homeomorphism classification of contractible topological 4-manifolds (possibly with boundary or noncompact)? In the compact case, any such manifold has a homology 3-sphere as its ...
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1answer
38 views

Self intersection and cohomology of boundary of tubular neighbourhood

Let $X$ a compact orientable manifold of dimension $2n$ and $Y$ a compact submanifold of dimension $n$. Further let $U$ a tubular neighbourhood of $Y$. When I did some calculations I got the ...
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2answers
33 views

Surjectivity of sending loop from fundamental group to the endpoint of its lift?

This question refers to the following map defined on a fundamental group of some topological space $X$ covered by a covering map $p:\bar{X}\rightarrow X$: $$r: \pi_{1}(X, x_{0}) \rightarrow p^{-1}(x_{...
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1answer
35 views

Lower bound of embedding dimension for finite CW-Complex of dimension $d$

Consider a finite CW-Complex $C$ of dimension $d$. Let $n$ be the smallest integer such that the complex embeds in $\mathbb{R}^n$. From Whitney it follows that $n \leq 2d$. How would you bound the ...
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0answers
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What is a coboundary or a cocycle in a simplicial complex?

I'm having trouble understanding cohomology as all the resources that I find are too abstract and maybe not confined to algebraic topology. My question is simple: on a simplicial complex, what does ...