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

learn more… | top users | synonyms (1)

2
votes
0answers
47 views

Octonionic Hopf bundle is a bundle.

We have the octonionic Hopf map $f: S^{15} \to S^8$, where $f$ is a bundle of fiber $S^7$. To me this is not obvious. How do I see that the octonionic Hopf bundle is indeed a bundle? Thanks.
0
votes
0answers
25 views

Acyclicity and connectivity

For a topological space $X$, set: $$\text{acyclicity} (X) =\max\{k : \tilde{H_i}(X,\mathbb{Z}_2) = 0,\text{ for every }\,\, 0\leq i\leq k\} $$ Is there an example of triangulable topological space ...
1
vote
1answer
36 views

Prove that $S$ is colorable if and only if it is orientable

I am taking a course on algebraic topology and I am trying to prove the following exercise: Let $S$ be a differentiable surface in $\mathbb{R}^3$. Prove that $S$ is colorable (you can paint one ...
1
vote
1answer
61 views

Local degree of a polynomial defined on the Riemann sphere at a root

I'm working on a problem from Hatcher's Algebraic Topology, and I want to show that if we take polynomial $f(x)$ definded on the Riemann sphere mapping to the Riemann sphere, the local degree of the ...
0
votes
1answer
55 views

What is the name of the $(k-1)$ faces of a $k$ cell?

Is there an own name of the $(k-1)$ cells that are attached to a given $k$ cell (or in other words: of the $(k-1)$ cells that intersects a given closed $k$ cell, or yet another words: of the $(k-1)$ ...
0
votes
0answers
39 views

Fundmental group of cone

Let $K$ be a simplicial complex with an edge loop $\alpha=(a_0,a_1,a_2)$. Let $X$ be a simplicial complex consisting of $K$ together with another additional vertex $b$ in the center, connected to ...
1
vote
2answers
42 views

if $(X,A)$ has homotopy extension, so does $(X \cup CA,CA)$

I guess I could use the property: The homotopy extension property for $(X,A)$ is equivalent to $X\times \{0\}\cup A\times I\ $ being a retract of $X\times I$. Then there is a retraction:$$X\times ...
3
votes
2answers
73 views

Why is this not a well-defined $\Delta$-complex of the torus?

My lectures notes say that the second diagram isn't a well-defined $\Delta$-complex of the torus because the $2$-simplices aren't totally ordered. I don't really understand what that means. Let's ...
0
votes
0answers
38 views

Does the word of an attachment map for some cell on a CW complex correspond to the cellular boundary map?

I'm in the middle of reading Hatcher, Chapter 2.2. The definition he gives for the boundary map for cellular homology of a CW complex $X$ is for some $n$-cell, $e^n_\alpha$, $d_n(e^n_\alpha) = ...
3
votes
1answer
54 views

Let $f:S^n\to S^n$ be continuous. Can the Image of An Open Set Under $f$ Have Non-Empty Interior?

Let $f:S^n\to S^n$ be a continuous map. Let $x\in S^n$ and $y=f(x)$. Assume that there is a neighborhood $U$ of $x$ in $S^n$ such that no element of $U-\{x\}$ maps to $y$. Question. Can the ...
6
votes
0answers
51 views

Proving that the given map has the path lifting property

Let $X$ be a locally compact metric space and $G$ be a discontinuous group of homeomorphisms of $X$. I need to show that the orbit map $p :$ $X \rightarrow X/G$ has the path lifting property. ...
1
vote
1answer
30 views

If $f(x) \neq g(x)$ for every $x ∈ S ^n$, then $g$ is homotopic to $a ◦ f$

Is this generalization that any map $f: S^n → S^n$ with no fixed points is homotopic to the antipodal map true? Let $f , g : S ^n → S^n$. Show that if $f(x) \neq g(x)$ for every $x ∈ S ^n$, then $g$ ...
1
vote
1answer
50 views

Prove that the group of homeomorphisms with the given metric is a complete metric space

Let $(X, \rho)$ be a Complete metric space, such that $\sup\limits_{x,x'\in X}\rho(x,x') < \infty$. Let $G=G(X)$ be the group of homeomorphisms of $X$ with itself and let ...
0
votes
0answers
24 views

Special Case of Universal Coefficient theorem

Show that if $C_* :$ $ ...\to C_2 \to C_1 \to C_0 \to 0$ be a complex of vector spaces over a field $k$, then $H^n$($Hom_k(C_*,k)) \cong$ $Hom_k(H_n(C_*),k)$.Does this result holds if $k$ is not a ...
2
votes
1answer
33 views

Connectivity and product

Let $X$ and $Y$ be two topological spaces. Is there any relation between Connectivity of $X\times Y$ and connectivity of $X$ and $Y$? I think by Kunneth formula, it must be $Conn(X\times ...
1
vote
1answer
109 views

Is there is a way to construct a covering space of a wedge of two circles for a given normal subgroup

I would like to construct a covering space of a wedge of two circles with a given normal subgroup $H \subset \pi_{1}(S^{1} \vee S^{1})=F_{a,b}$. The goal is to find a covering space $\tilde{X}$ so ...
5
votes
2answers
89 views

Formula relating Euler characteristics $\chi(A)$, $\chi(X)$, $\chi(Y)$, $\chi(Y \cup_f X)$ when $X$ and $Y$ are finite.

This is a followup to my question here. Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the ...
6
votes
1answer
63 views

Reference request: homotopy groups of $\mathbb{C}P^n$ in terms of homotopy groups of spheres?

Could anyone work out/supply a reference to the computation of homotopy groups of complex projective space $\mathbb{C}P^n$ in terms of the homotopy groups of spheres?
2
votes
1answer
39 views

Examples of finite simplicial sets

Let $K$ be a simplicial set. A simplex $x\in K_{n}$ is said to be non degenerate if it is not the degenerancy of a $n-1$ simplex, i.e if there is no $y\in K_{n-1}$ such that $s_{i}y=x$. A simplicial ...
0
votes
0answers
38 views

homotopy between constant simplicial sets

Assume that $K,L$ are constant simplicial sets (i.e all the faces maps and the degenerancies maps are equal to the identity and $K_{i}=K_{0}$, $L_{i}=L_{0}$ for $i>0$). Assume that there is a ...
3
votes
1answer
67 views

Diagram in Hatcher's Algebraic Topology - what do the arrows mean?

Here is a picture from Chapter 2 in Hatcher's Algebraic Topology: The context is that the space on the left does not have the structure of a $\Delta$-complex, whereas the shape on the right does. On ...
1
vote
1answer
48 views

Is a square a $\Delta$-complex?

Notation and definitions: $\Delta^n$ is the standard $n$-simplex with ordered vertices $[v_0,\ldots,v_n]$; $[v_0,\ldots,\hat{v_i},\ldots,v_n]$, where the hat denotes omission, is a face of ...
1
vote
0answers
64 views

obstruction to lifting a projective bundle to a vector bundle

Why is the obstruction, to lifting a projective bundle to a (complex) vector bundle on a space $X$, given by an element $\alpha \in H^3(X, \mathbb{Z})$? Vector bundles are classified by $H^1(X, ...
5
votes
1answer
58 views

If $G$ is a group of isometries of $X$ then prove that $X/G$ and $X/\bar{G}$ are homotopically equvalent

Let $X$ be a connected, locally path connected, locally compact metric space. Let $G$ be a group of isometries of $X$ (that is a group of homeomorphisms of $X$ with itself that preserves distance). ...
1
vote
2answers
101 views

Reference Request/Independent Study Algebraic Topology

Over my winter break, I am planning on trying to scratch the surface of algebraic topology. I am already comfortable with introductory/intermediate abstract algebra, and topology to the extent of ...
5
votes
0answers
76 views

Is $Y \cup_f X$ a CW complex?

Let $A$ be the subcomplex of a CW complex $X$, let $Y$ be a CW complex, let $f: A \to Y$ be a cellular map, and let $Y \cup_f X$ be the pushout of $f$ and the inclusion $A \to X$. My question is, is ...
0
votes
1answer
60 views

How to show $\mathbb R^n-\mathbb R^m$ is homotopy equivalent to $\mathbb R^{n-m}-\{0\}$

How to show $\mathbb R^n-\mathbb R^m$ is homotopy equivalent to $\mathbb R^{n-m}-\{0\}$? Can I find some homotopy map to prove this? I can imagine the picture, how can I find a real homotopy?
2
votes
0answers
80 views

Fundamental group of the complement in $\mathbb R^3$ of a line and a circle

What is the fundamental group of the complement in $\mathbb R^3$ of a line and a circle. There are actually two cases to consider, one where the line goes through the interior of the circle and the ...
3
votes
1answer
41 views

If p is a covering map of a connected space, does p evenly cover the whole space?

Suppose I have a covering map $p:X\rightarrow Y$, and $Y$ is connected. Is $Y$, as an open set, evenly covered by $p$? I think the answer is yes; I'm new to this kind of topology, so I'm not sure if ...
4
votes
1answer
82 views

The fundamental group of $\mathbb S^2$ attached with a diameter? [duplicate]

What is the fundamental group of $\mathbb S^2$ attached with a diameter? And what is the fundamental group of a hemisphere attached with a diameter? I guess for the latter one, we can deformation ...
6
votes
2answers
87 views

Generalized Jordan-Brower separation theorem

Suppose $M^{n+1}$ is a closed connected smooth manifold and $N^n$ is a closed connected smooth embedded submanifold of $M^{n+1}$. What's the weakest condition under which the Jordan-Brower ...
5
votes
1answer
49 views

What does “cst” stand for in algebraic topology?

This is a question on notation present in this post. Define $E_p = \{ (y, \gamma) \in E \times B^{[0,1]} \mid p(y) = \gamma(0) \}$. There's a map (in fact a fibration) $q : E_p \to B$, $(y,\gamma) ...
0
votes
1answer
65 views

Intuition on homotopy group

I just started learning some algebraic topology and trying to make sense of the following identity, $$ \pi_m(S^n) = \begin{cases} \mathbb{Z} , & m = n \\ 0 , & m < n \, . \end{cases} $$ I ...
2
votes
1answer
42 views

What, exactly, is a vertical homotopy?

As the question title suggests, what exactly is a vertical homotopy? Googling has failed to provide any results as so far as a clear definition goes...
4
votes
1answer
109 views

Homotopic vs. homologous simplices/chains

Just learning about simplicial homology. Suppose $X$ is a topological space, and suppose I have a $1$ simplex $\sigma_1 : \Delta^1 \to X$ which is itself a cycle, i.e., it descends to a map from ...
3
votes
0answers
31 views

Proof that $K^*(BG)=K^*(BT)^W$.

I was wondering if anyone had a refrence for the fact that $K^*(BG)=K^*(BT)^W$ for $G$ a compact connected lie group, $T$ a maximal torus, and $W$ the associated Lie group. I was able to derive this ...
2
votes
1answer
93 views

Intuition for the Dold-Kan correspondence

maybe this question does not make sense and it's just a psychological problem of mine. However I cannot understand the geometric picture of the Dold-Kan correspondence. Let $\mathbf{Ab}$ denotes the ...
2
votes
1answer
58 views

fundamental group of two circles which intersect at two common points

I am trying to solve this by Van-Kampen's theorem. What I do is just move a point at left side and a point at right side. Then I get two open sets whose intersection deformation retracts to a circle. ...
1
vote
0answers
115 views

First Cohomology group trivial for a simply connected manifold?

We know that the for a simply connected manifold X, both the first fundamental group and the first homology group are both trivial. Is it also true that the first cohomology group is also trivial? I ...
0
votes
1answer
59 views

How can a covering map (as defined in Hatcher) fail to be a surjection?

Hatcher defined a covering space as follows: $\textbf{Defn:}$ A covering space of a space $X$ is a space together with a map $p: \tilde{X} \to X$ satisfying the following conditions: There exists an ...
1
vote
0answers
58 views

Base change of topogical spaces VS Base change of schemes

In algebraic geometry, we have the following famous base change theorem [Hartshorne III Theorem 12.11]: Let $f:X\to Y$ be a projective morphism of noetherian schemes, and let $\mathcal F$ be a ...
1
vote
2answers
78 views

Simpler way of computing the first homology group of $\Delta^4$ (without the interior)?

Here's my attempt. The first part (computing $Z_1$) feels like it's far too complicated. First of all - is my work correct? And second, is there a simpler way of doing this? Let's label the ...
0
votes
1answer
61 views

Manifold with Atlases of different Dimension

Given a set M with a topology, will two different Atlases $A_1$ & $A_2$ have same dimension? Does the following argument hold : - If at a point $x$ i have a chart $(U,\phi)$ in $A_1$ $(U,\psi)$ ...
1
vote
1answer
25 views

How to see that an induced Outer automorphism does not depend on the choice of the Out-Inverse function?t

A marked graph is a pair $(G,τ)$ where $G$ is a graph and $τ: R_n \to G$ is a homotopy equivalence . $R_n$ is the Rose with n petals which is isomorphic to $F_n$ ,the free group with $n$ generators. ...
7
votes
1answer
130 views

Why care about the $(\infty, 1)$-category of topological spaces?

While learning about homotopy in my Algebraic Topology course I (as someone who is at least aware of higher category theory) noticed that it's possible to define a notion of "homotopy between ...
1
vote
1answer
61 views

Infinite connected sum of S^n

I am curious about the following statement : Statement : Infinite connected sum of $S^n$ is homeomorphic to $\mathbb{R}^n.$ Any hint, proof of reference will be appreciated. Thank you.
8
votes
2answers
83 views

Difference between homotopy equivalence and based homotopy equivalence?

As the question suggests, what is the difference between homotopy equivalence and based homotopy equivalence?
3
votes
0answers
58 views

$\mathbb{R}^3\setminus S^1$ deformation retracts to $S^2 \bigvee S^1$. How? [closed]

We know $\mathbb{R}^3\setminus S^1$ deformation retract of $D^3\setminus S^1$. It is ok. $D^3\setminus S^1$ can be thought like a solid torus and D^2. How can i imagine?
4
votes
0answers
57 views

Can a disconnected surface have a (negative) genus?

This question is rather about a convention. Is it possible (and conventional) to asign to, say the disconnected sum of two connected surfaces $X=\Sigma_h \sqcup \Sigma_g$ a genus? Since one has ...
3
votes
1answer
154 views

Is the orientation double cover unique?

My question comes from the following. We usually say "the universal cover" because, as we know, it is unique up to isomorphism on the adequate category (in particular, every universal cover is ...