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)

1
vote
1answer
45 views

Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
0
votes
0answers
10 views

If $i$ is an inclusion why is the induced $i_*$ an epimorphism

Given the following exact homology sequence of a pair. This is in Example 2 (page 134) from Munkres. This is where I am always stuck computing homology using exact sequence. I cannot grab the last ...
1
vote
1answer
33 views

Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...
0
votes
0answers
4 views

Generalised orientation for a $(B,f)$-structure: A map of $(B,f)$-structures induces an orientation

As the title suggests, I'm interested in proving the following claim (Kochman example 1 page 135): Consider the identity map $$\text{Id}\colon \mathfrak{B}\to \mathfrak{B}$$ where $\mathfrak{B}$ ...
1
vote
0answers
28 views

An example of infinite Betti number refers to G-covering, and the motivation of G-covering

I saw from a book that if G is infinite, then when considering G-covering: $p:X'\to X$, the P-th Betti number of $X'$ can be infinite. Can you give me an example of that. Also I am curious why we need ...
5
votes
1answer
69 views

Why do subvarieties correspond to Hodge classes?

Let $X$ be a smooth complex projective variety and define $$Hdg^k(X)=H^{2k}(X,\mathbb{Z})\cap H^{k,k}(X)$$ the group of integral $(k,k)$ cycles on $X$. Now it is a fact that we can associate to the ...
1
vote
2answers
19 views

The number of intersection points between a trivial loop and any other loop in the torus

In a torus, let $C_1$ be a trivial loop (contractible to a point) and let $C_2$ be any simple loop (trivial or meridian or longitude). What is the intersection number of $C_1$ and $C_2$. I think that ...
4
votes
0answers
21 views

when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it ...
1
vote
1answer
51 views

When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$.

Let $X$ be a connected CW complex. Let $X_n$ fit into a commutative postnikov diagram for $X$ and let the fibrations $K(\pi_n(X),n) \hookrightarrow X_n \xrightarrow{\mathscr p} X_{n-1}$ be given. ...
1
vote
0answers
40 views

Why is the fundamental group of the plane with two holes non-abelian?

I know $\pi_1(\mathbb{R}^2\setminus\{x,y\}) = \mathbb{Z}\ast\mathbb{Z} = \langle a,b\rangle$, but it's non-abelian-ness isn't obvious to me. Specifically, I draw a box and two points to represent ...
0
votes
1answer
38 views

Show $\mathbb{C}P^n$ is a $2n-$manifold [in singular homology theory]

There is a Theorem in the book that says: The space $\mathbb{C}P^n$ is CW complex of dimension $2n$. I wonder some questions: Is there any Theorem or result that if a space has CW complex ...
0
votes
2answers
37 views

To show that product $Z=X×Y$ in the product topology is a CW complex

I would like to prove the following: If $X,Y$ are CW complexes, and either $X$ or $Y$ is locally compact then the product $Z=X×Y$ in the product topology is a CW complex. (-see here) In order to ...
6
votes
1answer
610 views

Local homology group: a homeomorphism takes the boundary to the boundary

Let $X \subset \mathbb{R}^{n}$ be the subspace $\{ x_{1},...,x_{n} \mid x_{n}\geq 0 \}$, and let $Y$ is the subspace with $x_{n}=0$. Let $x\in Y$, calculate the local homology of $X$ at ...
2
votes
0answers
12 views

A spectrum $I$ is $E$-injective iff the map $i:I\rightarrow I\wedge E$ is an inclusion of a retract.

I was reading some notes on stable homotopy theory and I came across the statement in the title of this question. "Suppose $E$ is a ring spectrum, then $I$ is $E$-injective if and only if the ...
1
vote
0answers
27 views

Does an isogeny always define a covering map?

Consider a map $f: G_1 \to G_2$ between two topological groups. If $f$ is an isogeny when viewing $G_1,G_2$ as algebraic groups does $f$ always define a covering map when viewing $G_1,G_2$ as ...
1
vote
0answers
13 views

Homology of triples

A triple of topological spaces $(X, A, B)$ consists of a topological space $X$ and two subspaces $A, B$ with $B \subseteq A \subseteq X$. A map of triples $f \colon (X, A, B) \rightarrow (Y, G, H)$ ...
4
votes
1answer
213 views

Products of CW-complexes

I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
0
votes
0answers
25 views

Does the fundamental group ALWAYS act discretely on the universal cover? [on hold]

I need a sharp answer at least in case of manifolds. Please give references for theorems or lemmas that you cite. Thanks.
1
vote
1answer
22 views

Induced homomorphism of a covering space

How can I determine what's the induced homology homomorphism of a covering $S^{n} \rightarrow RP^{n}$? I suppose that a Hurewicz homomorpism would be pretty effective, but since I know nothing about ...
16
votes
2answers
868 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
13
votes
2answers
253 views

Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
2
votes
1answer
35 views

The Presentation Complex of $\mathbf Z\times \mathbf Z=\langle x, y|\ xyx^{-1}y^{-1}\rangle$ is the Torus

Let $G=\mathbf Z\times \mathbf Z$ and let $\langle x, y| \ xyx^{-1}y^{-1} \rangle$ be a presentation for $G$. In Example 1.46 of Hatcher's Algebraic Topology, the author mentions that the ...
3
votes
2answers
99 views

Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...
1
vote
1answer
45 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
4
votes
0answers
46 views

Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
1
vote
1answer
183 views

$\Delta$-complex structure of the cone and the suspension

I am going around in circles trying to answer the following question: Let $Y$ be a $\Delta$-complex. Describe a $\Delta$-complex structure of its cone $CY=(Y\times[0,1])/(Y\times\{0\})$ its ...
1
vote
1answer
21 views

Resources that explains “Cut and Glue” Technique for Delta Complex?

I am looking for any resources (book/online) that teaches and further elaborates on how the "cut and glue" technique works for $\Delta$-complexes. To be precise, I am looking for techniques and at ...
8
votes
1answer
387 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
5
votes
2answers
87 views

What is wrong with this proof that the identity map of $S^1$ is nullhomotopic?

I have read that the identity map of the unit circle $S^1$ is not nullhomotopic. In fact, I am very new to the subject, so I wonder what is wrong with the following reasoning (that seems to suggest ...
6
votes
1answer
70 views

Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form ...
4
votes
1answer
726 views

When is Quotient Map a Covering Map

Group $G$ acts on topological space $X$. Also, $x,x'\in X$ not in the same orbit of $G$ have open $U$, $U'$ such that $g(U)\cap U'=\varnothing$ for all $g\in G$. I have shown that $X/G$ is ...
0
votes
1answer
43 views

Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not ...
0
votes
0answers
22 views

Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
2
votes
1answer
65 views

“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
2
votes
1answer
49 views

Nonzero-homologous simple loop in Mobius band only winds once

I have a question as follows: Let $C$ be a closed curve in the Mobius band without self intersections. Prove that if $C$ is of non-zero homology, i.e., $C$ does not bound any face, then $C$ winds only ...
1
vote
0answers
21 views

Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
0
votes
0answers
11 views

deformation retraction as mapping cylinder

In Hatcher's Algebraic Topology, the mapping cylinder is defined as the quotient space of the disjoint union $(X\times I)\sqcup Y$ (where $I$ is the unit interval) of a continuous $f:X\to Y$, where ...
0
votes
2answers
31 views

Is this the projective plane or the Klein bottle? (Fundamental polygon)

I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle If we treat the top green and red arrows a single arrow then we ...
0
votes
1answer
37 views

Link between $\mathbb{Z}$ and the fundamental group's of 'common' topological spaces

I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below: Why is this the case? Is it is because they are all realted ...
1
vote
3answers
61 views

I don't get it, does “augmented chain complex” actually mean anything?

If I understand correctly, chain complexes make sense in any category enriched in the world of pointed sets. In practice, there's also a notion of an augmented chain complex, where we have an extra ...
1
vote
2answers
44 views

Cohomology of a mapping torus

How does the monodromy in a mapping torus $K_{\phi}$ affect the de Rham cohomology, if at all? Maybe this is naive, but I don't see how twisting one of the ends of $K\times I$ via the diffeomorphism ...
0
votes
1answer
35 views

Computing homology group using Mayer-Vietoris sequence

Suppose I am given an exact sequence: $$0\to G\xrightarrow{f} \mathbb{Z} \xrightarrow{g} \mathbb{Z} \xrightarrow{h} H\to 0 $$ where the first $\mathbb{Z}=H_3(A\cup B)$ and the second ...
0
votes
1answer
49 views

Homotopy of two circles contained in an open ball.

The following question is on my homework assignment and I have no idea how to even start answering it: Are any two distinct $S^1$ → $B(0,r)$ maps Homotopic? You can assume the circles are ...
0
votes
0answers
23 views

What does “carried by” a subcomplex mean?

Elements of Algebraic Topology by Munkres says the following on pg. 32 Given a $1$-chain $c$, it is homologous to a chain $c_3$ that is carried by the subcomplex $M$... What does carried by a ...
1
vote
1answer
21 views

Homotopy 'diagrams' for Klein bottle and projective plane

Background: I recently discovered that the complement to the circle and vertical axis shown below is homotopy equivalent to a torus Also complement to three infinite straight non-intersecting ...
2
votes
2answers
166 views

Complex projective line homeomorphic to $2$-sphere

Define an equivalence relation $\sim$ on $X={\bf C}^2\setminus \{(0,0)\}$ by $(x_1,y_1)\sim(x_2,y_2)$ if and only if there exists $t \in C\setminus\{0\}$ such that $(x_1,y_1)=(tx_2,ty_2)$ show that ...
0
votes
1answer
28 views

Representing Covering Spaces by PErmutations

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher. Hatcher starts by ...
7
votes
1answer
92 views

$S^3\times \Bbb CP^\infty$ is not homotopy equivalent to $\left(S^1\times \Bbb CP^\infty\right)\big/\left(S^1\times \{x_0\}\right)$

Both $S^3\times \Bbb CP^\infty$ and $\left(S^1\times \Bbb CP^\infty\right)\big/\left(S^1\times \{x_0\}\right)$ have cohomology ring isomorphic to $\Bbb Z[a]\otimes \Lambda[b]$ with $|a|=2$ and ...
2
votes
1answer
65 views

Show $H_{dR}^1(S^n)=0$ for $n>1$ without de-Rham Theorem, and some similar questions.

Question Without using de Rham's theorem, prove: (1) Show $H_{dR}^1(S^n)=0$ for $n>1$. (2) Use (1) to show $H_{dR}^1(RP^n)=0$ (3) a n-form $\Omega$ is exact on $S^n$ if and only if $$ ...
2
votes
2answers
260 views

If the action of a group $G$ on $\mathbb{R}$ is properly discontinuous then G is isomorph to $\mathbb{Z}$?

Let $G$ be a topological group, acts on a topological space $X$, such that the map $f: G \times X \rightarrow X:(g,x)\mapsto g*x$ is continuous. We say that this action is $properly\;discontinuous$ ...