Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
3
votes
1answer
77 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
70 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
66 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
24 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
38 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
26 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
58 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
43 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
46 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
24 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
71 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
70 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
79 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
61 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
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
1
vote
0answers
46 views
Is $Y$ open in $X\cup_f Y$?
Let $X,Y$ be topological spaces, $A\subset X$ - a subspace and $f:A\rightarrow Y$ - a continuous map. Then we can define
$X\cup_f Y = X\sqcup Y/\{a\sim f(a)\quad a\in A\}$
Then the composition ...
2
votes
1answer
37 views
Curvature form, tangent bundle and structural group.
Let $T\mathbb{C}P^n$ the tangent bundle over $\mathbb{C}P^n$. We have that the Chern classes are the coefficients of the characteristic polynomial of curvature form $\Omega$ of $T\mathbb{C}P^n$:
$$ ...
0
votes
0answers
23 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
89 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
45 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
71 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
96 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) ...
5
votes
1answer
56 views
What is 1st homology group of $X = \text{plane} - \bigcup_{n}\{(1/n, 0)\}\cup\{(0,0)\}$?
What is 1st homology group of $X = \text{plane} - \bigcup_{n}\{(1/n, 0)\}\cup\{(0,0)\}$?
My guess is since it is $\operatorname{Ab}(\Pi_1(X))$.
It is a subgroup of ...
1
vote
1answer
33 views
Are there $CW$-complexes not homeomorphic to $\mathbb{R}P^2$ but homotopy equivalent to $\mathbb{R}P^2$ with at most $5$ cells?
Is there a $CW$-complex $X$ not homeomorphic to $\mathbb{R}P^2$ but homotopy equivalent to $\mathbb{R}P^2$ with at most $5$ cells in its cell structure?
1
vote
1answer
41 views
Suppose one glues a mobius band to the boundary of a disk. What familiar space is this homeomorphic to?
I was doing a problem in algebraic topology and I need to gain knowledge of the following fact to procede. Suppose one glues a mobius band to the boundary of a disk. I want to calculate the ...
6
votes
4answers
67 views
What does it mean for a space to nontrivially cover itself?
I am going through qualifying exam questions and I came to a concept involving covering spaces of whose definition I did not understand.
What does it mean for a space to nontrivially cover itself?
...
2
votes
3answers
47 views
Klein Bottle discrete harmonics?
Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus.
...
5
votes
0answers
50 views
explicit cocycle representing Stiefel-Whitney class in Milnor and Stasheff
I am trying to do Problem 7A in Characteristic Classes by Milnor and Stasheff which asks the reader to do the following:
Identify explicitly the cocycle $C^r(G_n) \cong H^r(G_n)$ which ...
3
votes
1answer
63 views
Basic question about definition of Chern classes
Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)?
If we define chern ...
2
votes
2answers
31 views
All maps from a CW complex to S^1 null-homotopic implies finite first homology
So given a connected compact CW-complex $X$, a quick covering space argument shows that if $H_1(X)$ is finite, then every map $X \to S^1$ is null-homotopic. I was curious if the converse was true: ...
1
vote
0answers
30 views
Kernel of Hurewicz map using the spectral sequence of the universal cover
In Davis & Kirk's Lecture Notes in Algebraic Topology, Exercise 159 reads:
Use the spectral sequence of the universal cover to show that for a path-connected space $X$ the sequence ...
0
votes
3answers
61 views
Compute the singular homology groups of $S^1$
How do I go about computing the singular homology groups of $S^1$?
Anything to get me started or a full answer is appreciated.
EDIT: I've realised that while this is a simple question, there are ...
1
vote
1answer
75 views
Does every continuous map induce a homomorphism on fundamental groups?
Let $X$, $Y$ be topological spaces and $f:X \to Y$ be a continuous map. Does $f$ induce a homomorphism $f_* : \pi_1(X) \to \pi_1(Y)$? If not, what are the conditions on $f$ so that $f_*$ would be a ...
6
votes
0answers
78 views
When does a cohomology theory have a ring structure?
I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
7
votes
0answers
77 views
Existence of a map in a Hilbert space
Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$.
Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
4
votes
1answer
78 views
Is a subgroup of a fundamental group a fundamental group?
Let $(X,\ast)$ be a based topological space (maybe path connected or not, I don't know if this will be relevant to the solution).
Let $\pi:=\pi_1(X,\ast)$ be its fundamental group and let $H$ be any ...
4
votes
1answer
91 views
graphs and homotopy extension property
If $T$ is a spanning tree of a graph $X$. How to prove that the pair $(X,T)$ has the homotopy extension property, without using the definition of CW complexes? I mean I don't need the general case ...
