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

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1
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53 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
366 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 ...
4
votes
1answer
770 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
131 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 ...
4
votes
2answers
253 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 $i : A\times Y \rightarrow X\times Y$ ...
3
votes
2answers
91 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
128 views

Calculating the Picard group of $\mathbb{C}P^1$ in an elementary way

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 ...
9
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1answer
283 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 ...
5
votes
2answers
1k 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): ...
2
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2answers
118 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
vote
0answers
32 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
2answers
208 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
101 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
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0answers
113 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) = ...
2
votes
1answer
79 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
72 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
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0answers
96 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 ...
4
votes
2answers
655 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
106 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
351 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 ...
8
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0answers
211 views

Finite fundamental group in the Euclidean space [duplicate]

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
59 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 ...
5
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0answers
172 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. ...
5
votes
1answer
75 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
74 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?
3
votes
1answer
219 views

Suppose one glues a Möbius 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 Möbius band to the boundary of a disk. I want to calculate the ...
7
votes
4answers
103 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? ...
3
votes
3answers
127 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. ...
6
votes
1answer
274 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
155 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
79 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: ...
2
votes
0answers
113 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
584 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 ...
2
votes
1answer
328 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 ...
9
votes
1answer
168 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 ...
9
votes
1answer
244 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
2answers
485 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 ...
6
votes
1answer
193 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 ...
0
votes
0answers
185 views

Every compact, connected, orientable surface without boundary admits a $\Delta$ complex structure

Every compact, connected, orientable surface without boundary admits a $\Delta$ complex structure I'm not sure if the approach I'm using is even correct. But basically, I know that every compact ...
2
votes
1answer
140 views

Compact subset in a $\Delta$-complex structure

Suppose that $X$ is a topological space equipped with a delta-complex structure. Suppose that $K\subset X$ is compact. Prove that $K$ meets only finitely many open simplices. I managed to find a ...
7
votes
1answer
400 views

The action of the group of deck transformation on the higher homotopy groups

This is for homework. I'm supposed to do exercise 4.1.4 in Hatchers "Algebraic Topology", which is to show that given a universal covering $p: \tilde{X} \to X$ of a path-connected space $X$, the ...
2
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0answers
129 views

How to construct an $n$ sheeted covering space, $n \geq 2$, of a graph with trivial deck group.

How to construct an $n$ sheeted covering space, $n \geq 2$, of a finite connected graph with trivial deck group.
3
votes
1answer
165 views

Understanding homotopy types

I am currently self studying algebraic topology from the book "topology and groupoids". I don't understand classifying spaces up to homotopy type. By "I don't understand" I don't mean that I don't ...
2
votes
1answer
94 views

Problem on induced maps in cohomology

I am trying the solve the following problem: Let $g:\mathbb{C}P^\infty\longrightarrow \mathbb{C}P^\infty$ and suppose the induced homomorphism $$g^*:H^2(\mathbb{C}P^\infty)\longrightarrow ...
3
votes
1answer
410 views

Prove that a topological space equipped with a delta-complex structure is Hausdorff

Suppose that $X$ is a topological space equipped with a delta-complex structure. Prove that $X$ is Hausdorff. I can actually "see" why it should be Hausdorff after the hint from Hatcher asks to ...
2
votes
1answer
97 views

Quotient space about identity component

Let $T=\mathbb R/\mathbb Z$ be the circle group, $\mathscr A=C(T)$ be set of continuous function on $T$. $G(\mathscr A)$ denote set of invertible elements in $\mathscr A$, $G_{0}(\mathscr A)$ denote ...
5
votes
2answers
455 views

Winding number in higher dimensions

I am searching for references about the generalization in higher dimensions of the winding number (or "engulfing number") of a (hyper)surface $S$ around a point $p$, especially the identity of : (a) ...
2
votes
1answer
422 views

Van Kampen's Theorem with Torus and Projective Plane

I'm having some trouble finding the sets $U$ and $V$ to use for this problem. Let $T = S^1 \times S^1$ be the torus and $P=S^2/(x \sim -x)$ the projective plane. Form the space $X$ by identifying the ...
5
votes
1answer
149 views

Degree of maps on the 3-sphere

I am currently in the process of going through Ticciati's Quantum Field Theory for Mathematicians, which states the following (Theorem 13.7.11): "Let $g$ be a differentiable function from $S^3$ to a ...
3
votes
1answer
119 views

Classifying space of the reals

What's the classifying space $B\mathbb{R}$ of the additive group of real numbers provided with the Euclidean topology ? By the extension $\mathbb{Z} \hookrightarrow \mathbb{R} \twoheadrightarrow ...