Tagged Questions

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
1answer
44 views

Sufficient condition for a direct limit of abelian groups to be infinitely generated

I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
0
votes
1answer
28 views

Chern classes tangent bundle $\mathbb{C}P^n$.

Let $V \in Vect_k(M, \mathbb{C})$. We define Chern classes $c_i(V) \in H^{2i}(M, \mathbb{Z})$ with the usual 4 axioms. Now we consider the tangent bundle $$ \mathbb{C}^n \hookrightarrow T\mathbb{C}P^n ...
1
vote
2answers
49 views

Uniqueness of Seifert surfaces of knots

I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice ...
9
votes
1answer
83 views

Homology of $\mathbb{R}^2$ under the equivalence $x \sim 2x$

I was computing some examples of homologies of quotient spaces and I thought of the following. Does anyone know how to compute the homology groups of $\mathbb R^2/\sim$, where $\sim$ is the ...
4
votes
1answer
44 views

Constructing a odd homeomorphism between $A$ and $S^n$.

Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is homeomorphic to some sphere $S^n$, $n\leq N$ ($n$ is the dimension of the sphere). Is ...
0
votes
1answer
100 views

number of simplices in barycentric subdivision

Let $K$ be a simplicial complex. Is there a way to calculate the number of k-simlices in the barycentric subdivision $K'$ of $K$? Given the number of $l$-simplices in $K$, for any $l$, of course. (I ...
2
votes
1answer
51 views

Need help understanding statement of Van Kampen's Theorem and using it to compute the fundamental group of Projective Plane

I'm using the statement from Hatcher. I really don't understand the statement of the theorem, let alone the proof, and I especially don't understand what the normal subgroup $N$ generated by ...
1
vote
1answer
29 views

Action of fundamental group on n-th homotopy groups for RP^n

Is there any short way to see that the action of $\pi_{1}(RP^{n})$ on $\pi_{n}(RP^{n}) = \mathbb{Z}$ is trivial for $n$ odd and nontrivial for $n$ even? Maybe something without much machinery (smth ...
2
votes
1answer
61 views

How to compute the homology group $H_q(\mathbb{R}^n-e^r)$?

Let $e^r$ be a homeomorphic copy of $I^r$ in $\mathbb{R}^n$($I=[0,1]$).How to compute the homology group $H_q(\mathbb{R}^n-e^r)$?($r,n,q$ are non-negative integers)
2
votes
0answers
154 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
4
votes
2answers
56 views

Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex

I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
1
vote
0answers
35 views

Does group of deck tranformations acts transitively on each fibre if it acts traansitively on one fiber?

i am reading bredon "Topology and Geometry " It states that if we have a covering map p : X ->Y s.t. p(x) = y.X,Y are Hausdorff, path connected and locally path connected etc. I have 2 questions: ...
7
votes
0answers
64 views

Let $A$, $B$ be subsets of $S^n$, n≥2. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then…

Let $A$,$B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$. I've thought to do it by contradiction ...
1
vote
1answer
43 views

Is $r:S^1 \to \{x_0\}$ a retraction?

Definition: Let $A$ be a subspace of $X$ with an inclusion $i:A \to X$. Then $r:X \to A$ is called a retraction if $r \circ i = id_A$ that is $r(a)=a$ $\forall a \in A$. I read the question Is the ...
3
votes
1answer
48 views

Determining the embedding space:

I have seen a lot of discussion of alternate geometries for example on a sphere or hyperbolic saddle as opposed to a plane: Has anyone consider the notion of that plane or hyperbolic saddle itself ...
3
votes
1answer
82 views

Question on Good Pairs

$(X,A)$ is a good pair if $\exists V\subset X$ s.t. $V$ is a neighbourhood of $A$ that deformation retracts to $A$. Prove that if $(X,A)$ is a good pair, then $(X/A,A/A)$ is also a good pair. How ...
5
votes
1answer
74 views

Induced map on homology from a covering space isomorphism

Suppose $S^1 \times \mathbb{R}P^2$ covers some space. Why is it that any covering space isomorphism $h$ induces the identity map on $H_1$? I don't see how to prove this except maybe from looking at ...
5
votes
2answers
92 views

An intuitive idea about fundamental group of $\mathbb{RP}^2$

Someone can explain me with an example, what is the meaning why $\pi(\mathbb{RP}^2,x_0)$ is $\mathbb{Z}_2$? consider this quotient on the disk representing the situation: $\mathbb{RP}^2$ (sorry ...
0
votes
0answers
17 views

Form a space X by identifying the boundary of M with C by a homeomorphism. Compute all the homology groups of X. [duplicate]

Let T denote the torus S1×S1 and let M denote the Möbius band. Let C be a simple closed curve in T which bounds a 2-disk. Form a space X by identifying the boundary of M with C by a homeomorphism. ...
3
votes
0answers
69 views

Hawaiian Earring

Let $X=[0,1]$ and $A=\{0\}\cup\{\frac{1}{n}|n\in\mathbb Z\}$. Note that $(X,A)$ is not a good pair. Show that $H_1(X,A)$ is not isomorphic to $H_1(X/A)$. I have a sequence of homology groups: ...
2
votes
0answers
27 views

Prove that $s Sq^i =Sq^i s$.

I have been studying the mod 2 Steenrod algebra. And I try to solve some exercises of it. Can you help me to check this proof: Let $SX$ denote the suspension of $X$, and let $S: \underline{H}^q(X) ...
4
votes
1answer
65 views

Hatcher 2.2 Exercise 33

The following is a question from Hatcher's "Algebraic Topology": Let $X$ be a space such that $X$ is the union of $n$ open sets $A_i$ with the property that every intersection $A_{i_1}\cap \dots ...
2
votes
2answers
44 views

If two Lie Groups are homomorphic, does that mean that they are homeomorphic?

I am studying Lie groups, and I had a simple question If two Lie Groups are homomorphic, does that mean that they are homeomorphic? I appreciate any help. Thanks in advance.
4
votes
2answers
72 views

Prove rigorously that for two points $x, y \in M$, the spaces $M \ \backslash \{x\}$ and $M \ \backslash \{y\}$ are homeomorphic.

Let $M$ be a connected topological manifold. Prove rigorously that for two points $x, y \in M$, the spaces $M \ \backslash \{x\}$ and $M \ \backslash \{y\}$ are homeomorphic. I am not sure the best ...
1
vote
0answers
39 views

Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.

Let $T$ denote the torus $S^1\times S^1$ and let $M$ denote the Möbius band. Let $C$ be a simple closed curve in $T$ which bounds a 2-disk. Form a space $X$ by identifying the boundary of $M$ with $C$ ...
1
vote
0answers
29 views

problem related to the Mayer-Vietoris Sequence

Let $D_{k}$ be the surface obtained by removing k small disjoint open 2-discs from the unit disc $E^{2}$. Show that $D_{k}\simeq G_{k}$, the k-leaved rose. Let $M_{k}$ be the surface obtained by ...
2
votes
2answers
59 views

A long exact sequence of free Abelian group is the direct sum of very short exact sequences.

A long exact sequence of free Abelian group is the direct sum of very short exact sequences. The definition of short exact sequences doesn't seem to be very common from what I can see online: An ...
3
votes
0answers
61 views

How to calculate characteristic classes of tensor products?

I was given the following as an execrise: Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship: $$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$ It is ...
4
votes
1answer
36 views

Restriction of Covering Space

I'm studying for an exam, and got stuck on the following exercise: Find all two-sheeted covering spaces for $X =\mathbb{S}^1 \vee \mathbb{S}^1$. Label the two circles of $X$ by $a$ and $b$. Attach ...
2
votes
0answers
44 views

The product of a cofibration with an identity map is a cofibration

This is a problem from the book "modern classical homotopy theory" which I can't solve. Let $i : A \rightarrow X$ be a cofibration and $Y$ any space. Show that $i : A\times Y \rightarrow X\times Y$ ...
2
votes
2answers
47 views

free groups and bouquet of circles

For any free group $F$ generated by the set $S$, one can construct a graph specifically a bouquet of circles $X$ s.t $\pi_1(X)=F$. My question is: Does this mean free groups are isomorphic to a free ...
2
votes
1answer
57 views

$Pic(\mathbb{C}P^1)$

I have to explain the Picard group to some people that doesn't know the concept of sheaf. So is there a method to calculate $Pic(\mathbb{C}P^1)$ without sheaf theory? Is there a simple and easy proof ...
1
vote
0answers
25 views

how to compute the cohomology ring of grassmannian G(4,2)

I need to compute the ring of cohomologies over the integers of the complex grassmannian G(4,2). As I understand, one can use the Schubert cells and cellular homology to show that the homology ...
4
votes
1answer
72 views

Proof of the Ham-Sandwich theorem

I have doubts about the proof of the Ham-Sandwich theorem descibed on planetmath (http://planetmath.org/proofofhamsandwichtheorem) and wikipedia (http://en.wikipedia.org/wiki/Ham_sandwich_theorem): ...
1
vote
2answers
47 views

Closed sets in $R^2$ with $d(A,B)=0$ but $A\cap B=\emptyset$

Let $(X,d)$ be a metric space and $A$ and $B$ subsets of $X$. Define the distance $d(A,B)$ to be $d(A,B)=\inf\{d(p,q)\mid p\in A, q\in B\}$. Give an example of two closed subsets $A$ and $B$ of the ...
-1
votes
0answers
71 views

A question about homotopy equivalent [closed]

if $X$ is contractible, for any topological space $Y$ is the product $X\times Y$ homotopy equivalent to $X$.
1
vote
0answers
25 views

continuous discrete open map and topological dimension

Is there anyone who can help me to answer this question : Let $\Omega$ be an open bounded and connected set of $\mathbb{R}^n$. Let $A\subset \Omega$ be a closed set of Lebesgue measure zero and whose ...
3
votes
1answer
81 views

$\pi_1(X)$ finite, show $f:X \to S^1$ is nullhomotopic

Suppose that $\pi_1(X)$ is a finite group. Show that any map $f:X \to S^1$ is nullhomotopic. My attempt: Since $\pi_1(X)$ is finite and $\pi_1(S^1)=\mathbb{Z}$ torsion-free, then the induced ...
1
vote
0answers
43 views

Cohomology and 1-forms with compact support

I'm, having troubles with the following Let $U$ be a bounded open set in $\mathbb{R}^{2}$ such that $\mathbb{R}^{2}\setminus U$ has $n+1$ connected components. Prove that $\dim(H_c^{1}(U))=n$. I ...
4
votes
0answers
62 views

The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
1
vote
0answers
47 views

Is $Y$ open in $X\cup_f Y$?

Let $X,Y$ be topological spaces, $A\subset X$ - a subspace and $f:A\rightarrow Y$ - a continuous map. Then we can define $X\cup_f Y = X\sqcup Y/\{a\sim f(a)\quad a\in A\}$ Then the composition ...
2
votes
1answer
38 views

Curvature form, tangent bundle and structural group.

Let $T\mathbb{C}P^n$ the tangent bundle over $\mathbb{C}P^n$. We have that the Chern classes are the coefficients of the characteristic polynomial of curvature form $\Omega$ of $T\mathbb{C}P^n$: $$ ...
0
votes
0answers
24 views

Application of Kunneth formula to chain maps (Hatcher exercise)

I'm working on the following problem from Hatcher, which is in the Kunneth Formula section at the end of the cohomology chapter, and I'm having trouble figuring out where to start. Any direction would ...
3
votes
2answers
94 views

An alternative description of the first Stiefel-Whitney class

I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please? The first Stiefel-Whitney class of a vector ...
1
vote
1answer
37 views

Weak homotopy equivalence

I know a continuous map $f:X\to Y$ between topological spaces is a weak homotopic equivalence if it induces isomorphisms on the corresponding homotopy groups, but what kind of information do I get ...
2
votes
0answers
46 views

Chern classes tangent bundle.

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
7
votes
0answers
76 views

Finite fundamental group in the Euclidean space

Is there an example of a (path-connected) subspace of $\mathbb{R}^3$ which has a nontrivial finite fundamental group? If not, why?
4
votes
2answers
47 views

Using retraction for show that:

Let $f:\mathbb{S^2} \rightarrow \mathbb{R^2} \diagdown \{(0,0)\}$ a continuous application. Proof that there is $(x_0,y_0,z_0)\in \mathbb{S^2}$ such that $f(x_0,y_0,z_0)=\lambda(x_0,y_0)$ for some ...
4
votes
0answers
99 views

Show that $f$ is a homeomorphism of $X$ onto $f(X)$

I am having trouble on the following question. Some help would be much appreciated. Let $X$ be a Hausdorff space and let $f: X \rightarrow \mathbb{R}^n$ be a proper injective continuous function. ...
0
votes
0answers
35 views

Which of the following spaces nontrivially cover themselves?

I am having some difficulties with a qualifying exam question. I would appreciate if someone could give me a little help. Which of the following spaces nontrivially cover themselves? (a) $S^3$ (B) ...

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