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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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Functoriality of parallel transport of a Hurewicz connection on a fiber bundle

Let $A\overset{\alpha}{\rightarrow}B$ be a Hurewicz fibration. Any Hurewicz connection defines parallel transport along curves in the base. In general, such parallel transport maps $\alpha^{-1}(b)\to \...
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Prove that there is no retraction from $X$ to the image of a loop $\gamma(t)$

Let $X = S^1 \times D^2$, pick $x_0 \in X$. Suppose $\gamma$ is an embedded loop based at $x_0$ which represents an $n \in \mathbb{Z} = \pi_1(X)$ with $n \neq \pm1$. Prove that there is no retraction ...
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1answer
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Crushing a central dividing circle in $\Sigma_2$ to a point

Crushing a central separating circle in $\Sigma_2$ to a point So presentation of $\Sigma_2$ is $<x_1,y_1,x_2,y_2:[x_1,y_1][x_2,y_2]>$. I understand that crushing this separating circle to a ...
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Reference for homology of real projective space in a field.

I need a reference in a book for the computation of the homology of real projective space with coefficients in an arbitrary field. I do know how to do the computation, and I also found an online ...
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1answer
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Homology Represents topological Subspaces

Consider $X:= \mathbb{PC}^n $ the projective space. It is well known that the integral homology of $X$ is $H_i(X, \mathbb{Z}) = \mathbb{Z}$ is given by: $0 \leq i \leq 2n$ even, and $H_i(X, \mathbb{Z}...
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1answer
47 views

Show that $\bigoplus_{i \text{ even}}C_i=\bigoplus_{i\text{ odd}}C_i$

Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i<0$. Suppose that $C_i=0$ for all $i$ is sufficiently large and that for all $...
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1answer
38 views

ext sheaf and cohomology

Let $\mathcal{F}$ be a sheaf on $\mathbb{P}^{3}$ with $\mbox{dim}(\mathcal{F}) = 0$. It's true that cohomology $H^{i}(\mathbb{P}^{3}, ext^{3}(\mathcal{F}, \mathcal{O}_{\mathbb{P}^{3}})) = 0$ for $i = ...
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1answer
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Is it possible to find a torsion free space with non-trivial $\operatorname{Sq}^{2^n}$

Torsion free means integral cohomology is torsion free. When $n \leq 3$, such spaces are projective space $\mathbb{CP}^n$, $\mathbb{HP}^n$ and the Cayley plane. $\operatorname{Sq}^{2^n}$ takes the ...
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2answers
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Is there an orientable closed compact $3$-manifold such that its fundamntal group is $\mathbb{Z}$?

Is there an orientable closed compact $3$-manifold such that its fundamental group is $\mathbb{Z}$? How about $\mathbb{Z^2}$?
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1answer
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Universal covering that induces zero on homologies

Let $p:\tilde{X}\rightarrow X$ be the universal covering space such that $p_*$ is zero on all homologies of dimension greater than zero. Does this imply that $X$ is $K(\pi_1(X),1)$? Working with the ...
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1answer
35 views

Sanity check: self-homeomorphisms of a punctured torus is discrete?

I know that the self homeomorphisms of a closed torus is disconnected, with connected component the group of translations, and component group the mapping class group, isomorphic to $SL(2,\mathbb{Z})$ ...
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1answer
45 views

surjective map $S^n \rightarrow S^n$ of degree zero

Construct a surjective map $S^n \rightarrow S^n$ of degree zero for ah $n\ge 1$. I’ve been struggling with this exercise from hatcher. I know that if the map is not surjective then the degree is zero,...
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1answer
39 views

Euler characteristic of $\mathbb{R}$ and $\mathbb{R}^2$?

I´m trying to understand the value of the Euler characteristic of the real line and the real plane. I don´t know if it is defined, I think that it is for any topological space. So this could be ...
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1answer
43 views

Map of H-spaces inducing zero on homologies

If a map of $H$-spaces $f:X\rightarrow Y$ induces zero on the homology groups at dimensions greater than zero does it necessarily induce zero map on the homotopy groups? It is definitely true for $\...
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1answer
58 views

Path Connectedness of Simply Connected Space Minus a Point

Suppose that $X$ is a simply connected topological manifold of dimension at-least $2$. Fix a point $x \in X$ and define $\tilde{X}\triangleq X-\{x\}$. How can I prove that the $0^{th}$ signular ...
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1answer
30 views

Empty boundary of a p-chain

Definition: A $p$-cycle is a $p$-chain with empty boundary: suppose that $\phi$ is a group homomorphism from $C_{p}$ to $C_{p-1}$. Then,$\forall \alpha \in C_{p}, \phi(\alpha) = e_{+} \in C_{p-1}$. ...
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1answer
41 views

Contractable Metric Spaces Homeomorphic to Euclidean Space

Is there a characterization of all metric spaces which are homeomorphic to a contractable subset of Euclidean space? This question is cross-referenced here.
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Prove that there is an isomorphism $\phi_n:H_n(C_*)\to\bigoplus_{\alpha\in\Lambda}H_n(C_*^{\alpha}) $

Let $\Lambda $ be a fixed set. For each $\alpha\in\Lambda $ is $\{C_n^{\alpha}\}_{n\in\mathbb{Z}}$ a complex of chains with homomorphism border $\partial^{\alpha}$. Prove that there is an isomorphism ...
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0answers
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isomorphism for real vector bundles and complex vector bundles with inner product.

Let $B$ be paracompact. Call $\mathrm{Vect}_{\mathbb R}(B)$ the category of real vector bundles over $B$ and $\mathrm{Vect}_{\mathbb C,f}(B)$ be the category of complex vector bundles with an inner ...
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Homology of the maps between complex projective spaces

Suppose $m > n$, and let $f : \mathbb CP^m \to \mathbb CP^n $ be continuous, the claim is that the induced map between the homology(over $\mathbb Z$) is zero. I have no clue why this should be ...
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Prove that $\{C_n\}_{n\in\mathbb{Z}}$ is a chain complex with homomorphism border $\partial:=\bigoplus_{\alpha\in\Lambda}\partial^{\alpha} $.

Let $\Lambda $ be a fixed set. For each $\alpha\in\Lambda $ is $\{C_n^{\alpha}\}_{n\in\mathbb{Z}}$ a complex of chains with homomorphism border $\partial^{\alpha}$. For each $n\in\mathbb{Z}$ we ...
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1answer
30 views

Question about Conway Polynomials of oriented links

Hi I've got a few questions on Conway polynomials in preparation for an exam this Saturday that I don't know how to do: Let $L$ be an oriented link. (a) If $\mu(L)=1$ then $C(L)\in 1+z^2\mathbb{Z}[z]...
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1answer
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Question on linking number of weakly split links

Hi I don't have that many resources to learn this module and my exam is this Saturday. I'm having trouble proving the following: (a) Let $L$ be an oriented 2-component link of components $L_{1}, L_{2}...
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0answers
31 views

Local system associated to monodromy representation

How can I associate a local system to a representation $\rho: \pi_1(X) \to \mathbb C^*$? I have seen some construction, but it doesn't click for me. I know that the idea is to use a diagonal action ...
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0answers
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Is this an example of a local system

In Bott, Tu, one is asked to calculate the cohomology of the following sheaf: Exercise 10.7 (Cohomology with twisted coefficients). Let $\mathscr{F}$ be the presheaf on $S^1$ which associates to ...
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2answers
34 views

Homomorphism between $p$-chain groups

In the below paper, Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing the author provides an visual example of the boundary of a $p$-chain as seen ...
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0answers
30 views

geometric visualisation of an arbitrary p - chain

In the paper I am reading, the author has not provided a "methodology" for visualising what a $p$-chain looks like; a google search has not yield anything to assist me in this either. Persistent ...
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$f_n:C_n\to D_n$ is injective for all $n\in\mathbb{Z}$ but $f_*:H_n(C_*)\to H_n(D_*)$ is not injective for some $n\in \mathbb{Z}$.

Example of two complexes of chains $C_*$ and $D_*$ and a morphism of complexes of chains $f_*:C_*\to D_*$ in such a way that $f_n:C_n\to D_n$ is injective for all $n\in\mathbb{Z}$ but $f_*:H_n(C_*)\to ...
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Counting hypermaps with following properties

A $\textit{hypermap}$ of type $(g,n)$ is a graph embedded in an oriented surface of genus $g$ such that 1) the complement of the graph is the disjoint union of $n$ topological disks labelled from 1 ...
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On the equivalence of two different notions of the join of two topological spaces

I've seen two different notions of the join of two topological spaces $X$ and $Y$, namely: $X*Y := \left(X \times Y \times I \right)/ \sim$ where $\sim$ is the equivalence relation generated by $(x'...
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1answer
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Integral homology group of a 3-torus cut out a donut

I know that the integral homology group of the manifold $M$ is given by $$ H_j(M,\mathbb{Z}) $$ I also have tried that $H_j(T^3,\mathbb{Z})$ is given by $$ H_0(T^3,\mathbb{Z})=\mathbb{Z}, $$ $$ H_1(...
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Are there ways to combine manifolds that form groups? [on hold]

I'm thinking of group elements that might be labelled by 3-manifolds. So for example if $[m]$ is a group element. Then the group operation would be $[m]\times[n]=[p]$. For some manifolds $m,n,p$. ...
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1answer
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Determining the number of distinct p-chains

Definition: A $p$-chain is a subset of $p$-simplicies in a simplicial complex $K$. Let $T$ be a tetrahedron - a 3 - simplex. It is trivial to note that the faces of a tetrahedron are 2 - simplex; in ...
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4answers
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Whether $\Bbb{R}^3\setminus \{0\}$ and $\Bbb{R}^3 \setminus \{0,1 \}$ are homeomorphic or not?

I am thinking about whether the two spaces $\Bbb{R}^3 \setminus \{0 \}$ and $\Bbb{R}^3 \setminus \{0,1 \}$ are homeomorphic or not? I guess they are not homeomorphic but cannot find out the proper ...
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0answers
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Questions about topological properties of solution space of PDE

Consider nonlinear PDE whose global existence is not known while some other basic properties like a priori estimates, existence of weak solution and uniqueness of solution of certain regularity.(you ...
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1answer
63 views

Trying to understand the geometrization conjecture

The Thurston's geometrization conjecture says: Every closed orientable 3-manifold decomposes canonically into pieces whose interior has a locally homogeneous complete metric. I'm trying to ...
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0answers
63 views

Possible error in proof of PL-approximation (Hatcher)

I am reviewing some technical results from the fourth chapter of Hatcher's Algebraic Topology. In the proof of PL-approximation (Lemma 4.10), we let $B_1,B_2\subset e^k$ denote the balls of radius 1 ...
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Milnor's proof that a smooth manifold has the homotopy type of a CW complex

I have some questions about the proof of Theorem 3.5 of Milnor’s “Morse Theory”: At the end of the proof of this theorem, Milnor addresses the case when $f$ has infinitely many critical points: ...
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About deformation retract of pairs

Here: Relationship between homology of suspension of $X$ and $X$, Joe Moeller argued that $(\Sigma X\setminus U,C_-^n\setminus U)$ deformation retracts to $(C_+^n,X)$. I don't know the definition of ...
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There's no one-to-one continuous map from $\mathbb{R}^n$ to $\mathbb{R}^m$ for $n>m$? [duplicate]

There's no one-to-one continuous map from $\mathbb{R}^n$ to $\mathbb{R}^m$ for $n>m$ To do this problem, obviously I need to use the fact that $\mathbb{R}^n \cong \mathbb{R}^m$ iff $n=m$ by using ...
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$k^{th}$ homology of $n$-torus via Mayer–Vietoris sequence

Let $T^n$ denote $n$-torus defined as: $$ T^n = \underbrace{S^1\times S^1 \times \dots \times S^1}_n $$ The question is to compute $H_k(T^n)$ without using Kuneth formula. So far I've noticed that $...
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1answer
61 views

Proof of Kunneth theorem

What are different ways to prove Kunneth theorem relating singular homology of product space $X * Y$ in terms of homology of $X$ and $Y$? or reference?I know some ways: use cell homology for cell ...
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0answers
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$\mathbb{C}$ as an $ \mathbb{R}$-coalgebra

So I am trying to understand $\mathbb{C}$ as an $ \mathbb{R}$ coalgebra and I have already found out a comultiplication: $$\mathbb{C} \rightarrow \mathbb{C} \otimes_\mathbb{R} \mathbb{C} $$ $$(x+iy) \...
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1answer
41 views

Possible finite groups that can act properly discontinuously on $S^2$

Let $G$ act on $S^2$ such that for every $x\in X$ and every $g\in G, g\neq e$, there is a neighborhood $U$ of $x$ such that $g(U)\cap U\neq \varnothing$. If $S^2/G$ is a surface and $G$ is a finite ...
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1answer
29 views

Chain homotopy between homotopy equivalences [closed]

Are any two homotopy equivalences between two chain complexes always chain homotopic? I guess it’s not always true. What if they are chain complexes of free abelian groups.
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2answers
123 views

Fundamental group of a solid torus with a tunnel?

Imagine that you have a donut and a worm inside. This worm takes two turns around the solid torus, going back to the starting point after two laps. How could I find out what the fundamental group of ...
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0answers
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Homology groups of a Mobius strip attached to a Torus with a disc removed

Let $M$ be the mobius strip and $T$ be the torus. Suppose we remove the interior of a 2-disc, $int(D^2)$ from the torus and then we attach the mobius strip in a 2:1 mapping around the boundary circle ...
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1answer
56 views

What is the fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$

I want to know about fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ by Seifert-Van Kampen theorem. In my guessing, that is $\langle a_1, a_2 ,... a_n | a_1^{2}a_2^{2}\cdots a_n^{2}=1\rangle$. ...
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39 views

What does it mean to say $K$-theory satisfy Mayer–Vietoris sequence?

I saw the statement "a $K$-theory is expected to satisfy Mayer–Vietoris property and Bott Periodicity" somewhere and I am trying to understand what it means. What does it mean to say a $K$-theory ...
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1answer
46 views

Graph in genus 1

I want to understand the graph on the surface of genus 1. Let $G=(V,E,F)$ be a graph with $V,G,F$ denoting vertex, edge, and face respectively. Then the genus one condition give us that $V-E+F=0$, ...