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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Intuition of the meaning of homology groups
I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if ...
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1
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Properly discontinuous action: equivalent definitions
Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = ...
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Is the fundamental group of every subset of $\mathbb{R}^2$ torsion-free?
It seems that the fundamental group of any subset of $\mathbb{R}^2$ will not have an element of finite order.
Though the $3$-dimensional version is an open problem I couldn't immediately see why it is ...
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1
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Characterizing simply connected spaces
A topological space $X$ is simply connected if it is pathwise connected and each closed path $u : I = [0,1] \to X$ is path homotopic to the constant path at $x_0 = u(0) = u(1)$.
Recall that
A closed ...
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4
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If a covering map has a section, is it a $1$-fold cover?
If $q: E\rightarrow X$ is a covering map that has a section (i.e. $f: X\rightarrow E, q\circ f=Id_X$) does that imply that $E$ is a $1$-fold cover?
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If $h : Y \to X$ is a covering map and $Y$ is connected, then the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$.
In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition
Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (...
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CW complexes and manifolds
What is the strongest known theorem (if any) that classify which manifolds can be built as CW complexes?
Thank you guys. This is just a question I thought of during class, obviously not homework...
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9
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How can I prove formally that the projective plane is a Hausdorff space?
I want to prove the Hausdorff property of the projective space with this definition: the sphere $S^n$ with the antipodal points identified. It's seems easy, but I can't prove formally with this ...
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2
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Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group
Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.
I have tried all ...
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Homotopy equivalence of universal cover
As part of am exam question (Q21F here), I'm trying to prove that if $X$ and $Y$ are path-connected, locally path-connected spaces with universal covers $\widetilde{X}$ and $\widetilde{Y}$, ...
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Computing the homology and cohomology of connected sum
Suppose $M$ and $N$ are two connected oriented smooth manifolds of dimension $n$. Conventionally, people use $M\#N$ to denote the connecte sum of the two. (The connected sum is constructed from ...
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How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?
I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
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Compact subset in colimit of spaces
I found at the beginning of tom Dieck's Book the following (non proved) result
Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ ...
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Best Algebraic Topology book/Alternative to Allen Hatcher free book?
Allen Hatcher seems impossible and this is set as the course text?
So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.
...
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Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell
Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent.
I think it can be proved by showing they are both deformation retracts of $X\sqcup_H(...
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Why is every discrete subgroup of a Hausdorff group closed?
I have just begun to learn about topological group recently and is still not familiar with combining topology and group theory together.
I have read a useful property of discrete group on the ...
- 3,017
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2
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Tangent bundle of P^n and Euler exact sequence
I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has
$$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$
where $L$ ...
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3
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Suspension of a product - tricky homotopy equivalence
Let $(X,x_0), (Y,y_0)$ be well-pointed spaces (inclusion of the basepoints is a cofibration). Show the following homotopy equivalence
$$
\Sigma (X\times Y) \simeq \Sigma X \lor \Sigma Y \lor \Sigma (X\...
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29
votes
1
answer
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Triangulation of Torus
I was asked to find out the simplicial homology groups of the torus $T=S^1\times{}S^1$ embedded in $R^3$. I triangulated the torus like this :
Here the $0$-simplices are $\{v_0\}$. $1$-simplices are $...
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$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$
I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$.
At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
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2
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Why doesn't the "zig-zag" comb deformation retract onto a point, even though it's contractible?
I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6(c) in Chapter $0$. Unfortunately a picture is involved so it doesn't quite make sense for me to ...
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Which manifolds are parallelizable?
Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
- 6,196
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3
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the cone is contractible
Let $X$ be a topological space. I want to show that the cone $CX$ is contractible. Here we construct a deformation retraction from $CX$ to
the tip point of the cone
$$H_t: CX\to CX;\; (x,t')\mapsto (...
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votes
1
answer
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Every Group is a Fundamental Group
I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
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2
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Is homology determined by cohomology?
I am aware of the universal coefficients theorem for cohomology which implies that the homology groups completely determine the cohomology groups. I am wondering if cohomology determines homology in ...
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3
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connected manifolds are path connected
prove every connected manifold is path connected manifold .
my thought:
connected space : Let $ X$ be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint
nonempty ...
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votes
2
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composition of covering maps
The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering ...
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Examples of manifolds that are not boundaries
What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds?
Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional ...
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2
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Fundamental group of a torus with points removed
Question 5.33 from "Topology and its Applications" by Baesner is to compute the fundamental group of the torus ($T^2$) with $n$ points removed. I can "see" in my mind that if we remove one point we ...
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Fundamental group of complement of $n$ lines through the origin in $\mathbb{R}^3$
Just a quick question to verify whether I'm right.
Claim: The fundamental group of the complement of $n$ lines through the origin in $\mathbb{R}^3$ is $F_n$, the free group on $n$ generators.
Proof:...
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5
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Consequences of Degree Theory
I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
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$\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable
Question: Show that $\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable.
Motivation: This is one of those problems that I saw in Hatcher and felt I should be able to do, but couldn't quite ...
user2959
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4
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The circle bundle of $S^2$ and real projective space
Today I felt like computing the integral cohomology of the unit circle bundle of the tangent bundle of $S^2$. For completeness, it is defined by $SS^2=\{x\in TS\colon ||x||=1\}$, where we use the ...
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Equivalence of knots: ambient isotopy vs. homeomorphism
I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy:
These two knots are ambient ...
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2
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What is the homotopy type of the affine space in the Zariski topology..?
I'm asking this question out of curiosity, as I was unable to come to a conclusion.
Consider the affine space over an algebraically closed field, for concreteness we can work with $\mathbb{C}^{n}$. ...
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2
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Homeomorphism of the Disk
I'm working through Massey's "Basic Course in AT." One of the problems is prove that a homeomorphism of the closed disk maps the boundary to the boundary and the interior to the interior. How would ...
10
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composition of certain covering maps
This problem was posted before, but not the proof (because the asker knowed the answer), only a counterexample without the hypothesis of finite fibres. I want to know how to prove this proposition:
...
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Fundamental group of mapping torus?
Let $f\colon X\to X$ be a homeomorphism between a CW-complex $X$ and iteself.
Let $M_f=X\times [0,1]/(x,0)\sim (f(x),1)$, mapping torus of $X$ from $f$.
I want to calculate the fundamental group $\...
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Exercise 1.3.16 in Hatcher
I know this question has been asked before, but I couldn't find a satisfying answer so I hope to find it now.
Let $p:X\to Y$ and $q:Y\to Z$ be maps such that $q$ and $q\circ p$ are covering map and $...
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Show the quotient space of a finite collection of disjoint 2 simplices obtained by identifying pairs of edges is always a surface, locally homeomorp
Show the quotient space of a finite collection of disjoint 2-simplices obtained
by identifying pairs of edges is always a surface, locally homeomorphic to $\mathbb{R}^2$.
I have thought about doing ...
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1
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Intuitive Approach to de Rham Cohomology
The intuition behind homology may be summarized in a sentence: to find objects without boundary which are not the boundary of an object. This has geometric meaning and explains the algebraic boundary ...
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5
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Fundamental group of the special orthogonal group SO(n)
Question: What is the fundamental group of the special orthogonal group $SO(n)$, $n>2$?
Clarification: The answer usually given is: $\mathbb{Z}_2$. But I would like to see a proof of that and an ...
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An intuitive idea about fundamental group of $\mathbb{RP}^2$
Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$?
We consider the real projective plane as a quotient of the disk.
I didn't receive an ...
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how to compute the de Rham cohomology of the punctured plane just by Calculus?
I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without Mayer-...
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is the group of rational numbers the fundamental group of some space?
Which path connected space has fundamental group isomorphic to the group of rationals? More generally, is every group the fundamental group of a space?
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Unit sphere in $\mathbb{R}^\infty$ is contractible?
Let $\mathcal{T}_{\infty}= \left\{ U \subset \mathbb{R}^{\infty}: \ U \cap \mathbb{R}^n \in \mathcal{T}_n, \text{ for } n=1,2,... \right\} $.
Of course $\mathcal{T}_{\infty}$ is topology in $\mathbb{R}...
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map of arbitrary degree from compact oriented manifold into sphere
This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$.
I ...
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How to show Warsaw circle is non-contractible?
The Warsaw circle is defined as a subset of $\mathbb{R}^2$：
$$\left\{\left(x,\sin\frac{1}{x}\right): x\in\left(0,\frac{1}{2\pi}\right]\right\}\cup\left\{(0,y):-1\leq y\leq1\right\}\cup C\;,$$
where $C$...
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Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$? [duplicate]
Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
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Are $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ different?
I have seen the claim that $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ are not diffeomorphic, but I have only ever seen the proof that they are not isomorphic as vector bundles (which is a ...
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