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

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5
votes
0answers
175 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
244 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
107 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
131 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
287 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
158 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
80 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
122 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
656 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
344 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
179 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 ...
10
votes
1answer
249 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
514 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
195 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
191 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
151 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
433 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
votes
0answers
140 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
173 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
100 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
432 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
100 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
501 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
456 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
155 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
123 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 ...
1
vote
0answers
89 views

Canonical bundle and Möbius bundle

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
1answer
57 views

Isomorphism canonical and Moebius bundle.

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
4
votes
1answer
200 views

Groups acting on polytopes

I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs. Their basic ...
6
votes
0answers
180 views

Injective Resolutions in $\mathfrak{Ab}(X)$

Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
15
votes
1answer
206 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
10
votes
3answers
616 views

What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
3
votes
1answer
277 views

Strong deformation retract

Let $A$ be strong deformation retract of $X$ and $A=\alpha^{-1}(\{0\}) $ for some continuous $\alpha:X\to I$. If $H:X\times I\to X$ is homotopy between $i\circ r $ and $1_X$ (rel $A$) , where ...
-1
votes
2answers
49 views

An example about “rectract” but not “homotopy equivalence” [closed]

Can you help me to find an example for a pair $(X,A)$ such that $A$ is a retract of $X$, but $A$ and $X$ is not homotopy equivalent?
0
votes
0answers
71 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
2
votes
1answer
98 views

Van Kampen with complicated attaching map

I would like to know if it is possible to use van Kampen's theorem without knowing exactly what the intersecting space looks like. So given $U$ and $V$, and an attaching map, is it possible to work ...
8
votes
1answer
274 views

Triangulation of a manifold adapted to a submanifold

I am not extremely proficient in topology, and am concerned with the following question : Given a compact manifold $X$ and a submanifold $Y \subset X$, is it always possible to find a triangulation ...
5
votes
0answers
634 views

Orientability determined by top homology group

Let $M$ be a compact, connected $n$-manifold. Say that $M$ is orientable if there is a class $\alpha$ in $H_n(M)$ such that the reduced homology map $H_n(M)\rightarrow H_n(M,M\setminus\{p\})$ takes ...
0
votes
1answer
86 views

Universal bundles and classificant maps.

We have this theorem: Let $\xi=(E,p,X)$ be a vector fiber bundle with rank($\xi)=n$. We can find a map $f: X \rightarrow Gr_n(\mathbb{C}^{\infty})$ such that $\xi=f^*(\gamma_n)$. I denote with ...
5
votes
2answers
823 views

Conjugacy classes in the fundamental group

I want to solve the following problem (Hatcher Ch.1, problem 6): We can regard $π_1(X,x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S_1, s_0)→(X,x_0)$. Let $[S_1,X]$ be the ...
3
votes
0answers
127 views

Number of composition cycles on honeycomb graph embedded on torus

I am reading a lecture about dimers by R.Kenyon (http://arxiv.org/pdf/0910.3129v1.pdf). I have a question concerned to the honeycomb graph $H_n$ embedded on a torus (see the picture at page 24 of the ...
3
votes
2answers
87 views

Given a topological space $X$, does $H_1(X)=\mathbb{Z}$ imply $\pi(X)=\mathbb{Z}$?

Let $X$ be any topological space with the first homology group $H_1(X)=\mathbb{Z}$ . I claim $\pi_1(X)=\mathbb{Z}$. By Hurewicz, we know that $H_1(X)$ is the abelianization of $\pi_1(X)$. ...
2
votes
0answers
319 views

Chain map of isomorphic chain complexes

If we know that the homology groups of two chain complex are isomorphic to each other, can we say that there is a chain map between these two chain complexes? If so, how can we define the chain map?
4
votes
1answer
39 views

boundary map in the (M-V) sequence

Let $K\subset S^3$ be a knot, $N(K)$ be a tubular neighborhood of $K$ in $S^3$, $M_K$ to be the exterior of $K$ in $S^3$, i.e., $M_K=S^3-\text{interior of }{N(K)}$. Now, it is clear that $\partial ...
2
votes
0answers
175 views

Fundamental group of two 2-spheres attached at a point

Just wanted to confirm if my proof is correct and complete, trying to learn Van-Kampen Theorem. Question: Find the fundamental group of two copies of $S^2$ attached at a point . Proof: We claim that ...
2
votes
0answers
87 views

Describe two non-homeomorphic connected irregular covering spaces of $S^1\vee S^1$

I am having trouble with the following qual problem. If someone could help me get started it would be great. Describe two non-homeomorphic connected irregular covering spaces of $S^1\vee S^1$, each ...
2
votes
1answer
408 views

Is the product of two contractible spaces contractible?

If $X$ and $Y$ are contractible spaces, is $X\times Y$ a contractible space?
3
votes
2answers
715 views

induced map homology example

I am having trouble understanding how to compute the induced map for the second homology. For example say I have $\varphi:\mathbb{T}^2\rightarrow \mathbb{T}^2$ that is a self homeomorphism, then what ...