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

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3
votes
2answers
85 views

Example of a domain in R^3, with trivial first homology but nontrivial fundamental group

Let $\Omega \subset \mathbb{R}^3$ be a domain. Is it true that if $H_1(\Omega)$ = 0, then $\pi_1(\Omega) = 0$? For a counterexample, the group $\pi_1(\Omega)$ needs to be a perfect group and so I was ...
2
votes
1answer
38 views

Multiplication on a K(G,n)

Suppose that, given an abelian group $G$, there is a multiplication map $\mu:K(G,n)\times K(G,n) \to K(G,n)$ defined such that the induced map on the homotopy group $\mu_*:\pi_n(K(G,n) \times K(G,n)) ...
0
votes
0answers
53 views

Intersection form and poincaré duality

Let $ M $ be a $2n$-dimensional compact connected oriented smooth manifold and let $A$, $B$ be two $n$-dimensional submanifolds that intersect transversally. Denote by $A \cdot B$ the sum of the ...
1
vote
2answers
83 views

fundamental groups of open subsets of the plane

This should be a very basic algebraic topology question. The other day I was thinking about the fact that $P^2(R)$ has $\pi_1 = Z/2Z$. On the other hand I thought to myself how something like this ...
-1
votes
1answer
26 views

continuous maps between two compact CW complexes

Consider a continuous map $f: X \rightarrow Y$, where $X$ and $Y$ are compact $CW$ complexes. Are there interesting examples of $f$ such that the pre-image of a cell in $X$ is not a cell in $Y$?
4
votes
2answers
71 views

If the function $\varphi \colon Z\rightarrow C(X,Y)$ is continuous then $F\colon Z\times X\rightarrow Y$, $F(z,x)=\varphi (z)(x)$ will be continuous.

If the function $\varphi :Z\rightarrow C(X,Y)$ ($C(X,Y)$ with compact-open topology) is continuous and $X$ is locally compact, then $$F\colon Z\times X\rightarrow Y$$ $$F(z,x)=\varphi (z)(x)$$ will be ...
7
votes
2answers
136 views

Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
2
votes
1answer
106 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
3
votes
0answers
33 views

Is a sufficiently nice simple curve which is nulhomotopic the boundary of a surface?

This is a follow up to Is a simple curve which is nulhomotopic the boundary of a surface?. There, I asked whether, given a simple curve $C$ in an open subset $U$ of $\mathbb R^3$ which is nulhomotopic ...
0
votes
3answers
100 views

Top Cohomology of $\mathbb{P}^2$ via Sphere

I am trying to use the cohomology of the sphere to calculate $H^2(\mathbb{P}^2)$. My professor just mentioned there's an argument using the projection $\pi: \mathbb{S}^2 \to \mathbb{P}^2$ and the ...
4
votes
0answers
40 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
1
vote
1answer
37 views

Is a simple curve which is nulhomotopic the boundary of a surface?

Let $C$ be a simple curve in an open subset $U$ of $\mathbb R^3$. Suppose that $C$ is nulhomotopic in $U$. Must there exist a homeomorphism $f$ from the closed unit disk $D$ in $\mathbb R^2$ to $U$ ...
0
votes
1answer
101 views

Homeomorphism with a bouquet of two circles

I understand why removing two points from $\mathbb R^2$ gives a surface that is homeomorphic to a bouquet of two circles. But can someone please write this homeomorphism?
1
vote
1answer
118 views

Cohomology group of a torus with g holes

I have to compute the cohomology groups of a torus with g holes (the Riemann surface of genus g). first I have computed the cohomology of a Torus with 3 holes in the following way: I pick a covering ...
1
vote
1answer
72 views

Lefschetz Hyperplane Theorem: reference request

I've just begun working on my bachelor thesis on the "Lefschetz Theorem on Hyperplane Sections" (see for example http://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem). The goal of the thesis is ...
4
votes
0answers
43 views

what can you say about the degree of $f:\mathbb{C}P^n \to \mathbb{C}P^n$

Any thoughts on this problem: If $M$ and $N$ are simply-connected, $n$-dimensional manifolds, then $H^n(M;\mathbb{Z}) \cong \mathbb{Z} \cong H^n(N;\mathbb{Z})$. A map $f:M \to N$ induces a map ...
0
votes
0answers
51 views

additive isomorphism from $H^*(S^2 \times S^4 ; \mathbb{Z})$ to $H^*(\mathbb{C}P^3 ; \mathbb(Z))$

I need help in solving this problem: show that there is an additive isomorphism from $H^*(S^2 \times S^4 ; \mathbb{Z})$ to $H^*(\mathbb{C}P^3 ; \mathbb(Z))$. Then determine whether or not $S^2 \times ...
2
votes
0answers
32 views

isomorphic cohomology rings for the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times *)$ and $(S^3 \times \mathbb{C}P^{\infty})$

I need to show that the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times *)$ and $(S^3 \times \mathbb{C}P^{\infty})$ have isomorphic cohomology rings. any ideas ..... thanx!
2
votes
2answers
68 views

Euler characteristic of a Y-shaped pipe?

I'm familiar with the idea in topology that shapes that can be continuously deformed into one another are considered "equivalent". I read about the Euler Characteristic as being Vertices-Edges+Faces. ...
3
votes
1answer
372 views

Simplicial Complexes, Triangulation general question.

I am taking a first course in topology, and I am struggling with simplicial complexes. Specifically the triangulation of subspaces of $ \mathbb{R}^n $ confuses me. If you could help me on the ...
1
vote
1answer
38 views

What does $C_n(X)$ in a simplicial complex really mean?

My Algebraic Topology textbook says: Let $X$ be a simplicial complex. The group $C_n(X)$ of oriented $n$-chains of $X$ is the free abelian group generated by the oriented $n$-simplexes of $X$. ...
1
vote
1answer
180 views

Mobius Strip Cylinder

I am having trouble seeing why for the the regular cylinder $C=\{(x,y,z)|x^2+y^2=1,|z|\le 1\}$, $C/\mathbb{Z}_2$ is homeomorphic to the mobius band ($x_0 \in C$). Can someone explain?
1
vote
1answer
67 views

Determining the generators of cohomology (as a ring)

I am working on a problem to show that the cohomology graded rings of $\mathbb{C}P^3$ and $S^2$ x $S^4$ are not isomorphic (unreduced with integer coefficients) I have already calculated the graded ...
3
votes
1answer
79 views

n-truncated simplicial set

It might be a trivial question. So, I apologise in advance. Let $ \Delta ^{op}_n $ be the full subcategory of $ \Delta ^{op} $ such that the set of objects of $ \Delta ^{op}_n $ is $ \left\{ 0, ...
2
votes
0answers
45 views

Find Group Action

I'm asked to find a group action G on the unit cylinder C such that $C/G$ is homeomorphic to the torus. Would $\pm1 \cdot (x,y,z) = (x,y,\pm z)$ work? The only problem here is that G should only act ...
1
vote
1answer
60 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
2
votes
1answer
57 views

What is a loop in $\mathbb RP^n$

What is a loop in $\mathbb RP^n$ ? I have to show that: Given a loop $\alpha:[0,1]\rightarrow\mathbb RP^n$ starting and ending in $x_0=[N]=[S]$ and its lift $\tilde{\alpha}:[0,1]\rightarrow ...
5
votes
2answers
112 views

Is there an analogue of the universal cover for higher homotopy groups?

The universal cover $U$ of a topological space $X$ is a simply-connected covering space of $X$. As the 'universal' moniker implies, this space is universal in the category of covering spaces of $X$ ...
4
votes
3answers
211 views

Why is this a wrong Triangulation?

Why is this a wrong Triangulation ? I have to say, we had triangulation at the end of the topology course, so not in details. And the professor only mentioned the basic rules for the ...
0
votes
1answer
102 views

Showing that two spaces are homotopy equivalent

Let $x_0 \in S^1 \times S^1$. I want to show that $(S^1 \times S^1) - \{x_0\}$ and $S^1 \vee S^1$ are homotopy equivalent. We have to show that $\exists$ maps $f: X \rightarrow Y$ and $g: Y ...
2
votes
2answers
234 views

How to prove “Homotopy is an Equivalence Relation”

Reading Allen Hatchers book (available online via this link) on Algebraic Topology, it states on page 3 that homotopy type defines an equivalence relation. The symmetry and reflexiveness are ...
1
vote
2answers
49 views

Relation between closed 1-chain and closed paths

Let $\gamma\in Z_1X=\mathrm{kernal}\,\partial_{1}$, i.e., a closed 1-chain. Prove there exists a 1-chain $\delta=\sum_in_i\delta_i$, where each $\delta_i$ is a closed path, such that $\gamma-\delta\in ...
1
vote
1answer
56 views

Covering spaces and homotopical equivalence

I have this simple question: if $X$ and $Y$ are two topological spaces homotopically equivalents, have they the "same" covering spaces? (and if yes, in which sense?) This question derive from an ...
1
vote
1answer
97 views

$K$-theory exact sequence.

Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence $$ 0 \to ...
5
votes
3answers
203 views

Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...
5
votes
1answer
257 views

Hatcher Problem 2.2.36

I am struggling with the following question (2.2.36) from Hatcher for quite some time now: Show that $H_i(X\times S^n) \simeq H_i(X) \oplus H_{i-n}(X)$. I don't know how to use the hint given by ...
0
votes
1answer
23 views

Why $Sym^{n}(\mathbb{C})=\mathbb{C}^{n}$ or $Sym^{n}([0,1])=\Delta^{n}$ is a $n$-simplex?

Why $Sym^{n}(\mathbb{C})=\mathbb{C}^{n}$ or $Sym^{n}([0,1])=\Delta^{n}$ is a $n$-simplex. Where I can find theory about symmetric products?.($Sym^n(X):=X^n/S_n$ where $S_n$ is the symmetric group in ...
9
votes
0answers
125 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
3
votes
1answer
152 views

How to visualize projective plane

I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me? Moreover I want to understand ...
3
votes
0answers
57 views

The groups $[\Sigma^nX,Y]$ versus the homotopy groups

If $X,Y$ are pointed spaces, denote by $[X,Y]$ the pointed homotopy classes of pointed maps $X\to Y$. The sets $[\Sigma^nX,Y]$ actually have the structure of a group for $n\geq 1$. Here $\Sigma$ ...
1
vote
1answer
89 views

Prove fundamental group is the direct product

Suppose that $A$ is a retract of $X$ with retraction $r : X \rightarrow A$. Also suppose that $i_*(\pi(A,a))$ is a normal subgroup of $\pi(X,a)$. Prove that $\pi(X,a)$ is the direct product of the ...
5
votes
0answers
99 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
1
vote
0answers
72 views

prove that after removing countable infinite point from $S^{2}$,it will remain path-connected.

prove that after removing countable infinite point from $S^{2}$,it will remain path-connected. it was the question that arise in the algebraic topology course where I have this term.I thought ...
5
votes
3answers
363 views

Fundamental group of Hawaiian earring

I am trying to understand how the fundamental group of the infinite shrinking wedge of circles is $G=\prod_{i=1}^\infty\mathbb{Z}$. I understand that it is something more than ...
0
votes
1answer
27 views

Regarding an arbitary fibration as an inclusion

I'm reading through Allen Hatcher's Algebraic Topology, and he mentions that, given a Postnikov tower, the fibration $X_n \rightarrow X_{n-1}$, where $X_n$ and $X_{n-1}$ are CW complexes, can be ...
2
votes
0answers
56 views

Double coloring to distinguish mirror images

Recently, there was an interesting blog about distinguishing the right-handed trefoil from the left-handed trefoil using a variant of tricolorability (found in the following link): ...
3
votes
1answer
55 views

Trivial Cohomology Group->Lower-Dimensional Homotopy?

Calculating the (de-Rham) cohomology of a tee connector (Picture), I got $H^0=R,H^1=R^2,H^2=0$. Furthermore, just from looking at it, I assume the tee connector is homotopic to a circle with an arc ...
0
votes
1answer
42 views

Show that they are not the boundaries of any disjointly embedded disks.

This is an exercise in Hatcher's topology book. It's in Page 176, problem 4(b). In the unit sphere $S^{p+q-1}$,let $S^{p-1}$ and $S^{q-1}$ be the subspheres consisting of points whose last $q$ and ...
1
vote
1answer
76 views

Definition of the relative boundary map

According to Hatcher (page 115), since the boundary map $\partial: C_n(X)\rightarrow C_{n-1}(X)$ takes $ C_n(A)$ to $ C_{n-1}(A)$, it induces a quotient boundary map. I am trying to reformulate this ...
8
votes
1answer
109 views

What are necessary and sufficient conditions for the product of spheres to be paralellizable?

Okay, so I found the result that the tangent-bundle of any product of spheres is parallizable, given that some element of the product is either $S^1$, $S^3$, or $S^7$. I prove this as follows, first ...