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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3
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1answer
32 views

$H_q(X;\mathbb{Z})=0$ when X spherical complex with $H_q(X;F)=0$ for all $q>0$ and for all $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$

Suppose that X is a spherical complex with $H_q(X;F)=0$ for all $q>0$ where $F=\mathbb{Q}$ or $F=\mathbb{Z}/p\mathbb{Z}$, as $p$ runs over all primes. I want to prove $H_q(X;\mathbb{Z})=0$. I know ...
1
vote
0answers
246 views

chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
2
votes
1answer
143 views

what is th homology group of $\mathbb{Q}$?

what is the 0'th homology group of $\mathbb{Q}$ I mean $ H_{0}(\mathbb{Q})$?as the 0'th homology group is counting the path component of the space so it should be infinite direct sum of copies of $\...
2
votes
1answer
340 views

the fundamental group of punctured surface

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots [...
2
votes
1answer
56 views

Degree of a restriction of a continuous map?

I have a map $f:D^2 \rightarrow S^2$ and $f(-x)=-f(x)$ for $x \in S^1$. Does this mean that $\deg(f|_{S^1})=0$? if so, why? We defined this degree on $S^1$ as $f(\exp(t))=\exp(F(t))$ then $\deg(f)=F(1)...
1
vote
2answers
117 views

Fundamental groups and some properties

I have some basic questions about fundamental groups that came up when I tried to prove a few things: I am sorry that they are kind of informal questions, but I could not find any answers to them in ...
4
votes
0answers
110 views

Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
-1
votes
1answer
87 views

Associative property of free product of groups

I am reading Algebraic Topology by Allen Hatcher (available online at http://www.math.cornell.edu/~hatcher/AT/AT.pdf) and at line 1 of page 42, it reads: "... because of the relation $L_{gg'}=L_gL_g'$...
0
votes
0answers
124 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
-1
votes
1answer
72 views

Does there exist a “Möbius cloth”?

If such a cloth exists, then one should no more worry about the orientation of our clothes, which troubled me sometimes. :P Thus I am wondering Does there exist a non-orientable surface with $3$ ...
0
votes
2answers
246 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
0
votes
1answer
66 views

Questions about van Kampen's theorem.

I just read some things about van Kampen's theorem that threat this one from a different perspective than we discussed in class and this brought up a few questions: It was said that the images of the ...
5
votes
1answer
124 views

How do you construct the lifted topology of a groupoid cover?

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...
0
votes
1answer
80 views

Determine if these spaces are connected, Hausdorff, or compact.

Let $X = [0,1]/(0,1)$ and let $\pi: [0,1] \rightarrow X$ be the quotient map. Answer the following questions, proving your assertions: a) Is $X$ contractible? We need $s:X \rightarrow \{pt\}$ ...
0
votes
0answers
56 views

Understanding definition of properly discontinuous action

From Bredon, we say that the $G$-action on a space $X$ is properly discontinuous if "Each point $x \in X$ has a neighborhood $U$ such that $g(U) \cap U \neq \emptyset$" implies "$g = e$, an ...
3
votes
1answer
158 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
1
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0answers
112 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
1
vote
2answers
68 views

boundary of $M \times I$ where $M$ is the Möbius band

Let $M$ be the Möbius band and $I$ be the closed interval $[0,1]$. What is the boundary of $M \times I$? Is it orientable? What can I do when I want to know the boundary of such space? Please give an ...
1
vote
2answers
231 views

How to compute Euler characteristic from polygonal presentation?

How can I compute the Euler characteristic of a compact surface from its polygonal presentation $\langle S | W_1 , \ldots , W_k \rangle$? I guess that the number of edges is the number of different ...
7
votes
1answer
95 views

A correspondence between generators of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ and eq. classes of orthonormal frames

The problem is about (topological) orientation of $\mathbb{R}^n$: Define an equivalence relation on orthonormal frames in $\mathbb{R}^n$ by declaring two frames equivalent if the matrix expressing ...
4
votes
1answer
86 views

an arc is not a retraction of the Klein bottle

I want to use homology to solve the following problem: Prove that the circle represented by the blue arc in the picture is not a retract of the Klein bottle. (See the attached picture of the Klein ...
2
votes
1answer
67 views

The term “Homotopy” was given by whom?

I want to know the names who define the term 'Homotopy' in algebraic topology in 1907. Are they Dehn and Paul Heegaard? What is the full name of Dehn? Thank you in advance.
2
votes
0answers
48 views

Question about proof of van Kampens theorem

I found this proof of van Kampen's therem on page 2. link I don't understand the part where text 'since this is simply connected' occurs. Why does it follow that if $H_{i,0} $ is in $U_2$ that it is ...
3
votes
0answers
82 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary $\partial{\...
1
vote
0answers
161 views

Why is this not a triangulation of the torus?

I refer to example 4, fig.3.6, p.17 of Munkres' Algebraic Topology. He says the given triangulation scheme "does more than paste opposite edges together". Not clear to me. For those who don't have ...
3
votes
1answer
50 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of connected/compact/contractible/simply-...
4
votes
3answers
167 views

covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
3
votes
1answer
248 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
2
votes
1answer
236 views

mapping torus eqivalent definition

Let $X$ be a topological space and $f:X\to X$ a homeomorphism. I need to find a continuous, properly discontinuous $\mathbb{Z}$-action on $X\times\mathbb{R}$, such that the quotient $(X\times\mathbb{R}...
1
vote
0answers
73 views

Cohomology of Hawaiian earring?

Do the infinite wedge of circles and the Hawaiian earring have the same cohomology? I am happy that they have different homologies (the first is countably generated, the second uncountably).
0
votes
1answer
54 views

CW complex with no cells in dimension $n$

Hi need some help with the following problem: if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group. thanx.
7
votes
0answers
155 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object $...
1
vote
1answer
61 views

Is $S^2/\sim$ a $CW$-complex?

Consider the equivalence relation on $S^2$ define by $x\sim -x$ if $x\in S^{1}$ (we are supposed to see $S^1\hookrightarrow S^2$ as the equator) and $x\sim x$ otherwise. I have some questions ...
1
vote
1answer
74 views

Decomposition of cohomology group on $S^{n}$

If we have decomposition of cohomology group on $S^{n}$ it looks like $H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}$, where $H^{n}(S^{n})_{\pm}$ cohomology of invariant or anti-invariant $n$ ...
2
votes
1answer
85 views

Good resource for learning braid theory?

I recently heard about braid theory and read the Wikipedia article on it, and it seems really beautiful. What is a good resource for learning more about it? I have a background in mathematics at the ...
11
votes
1answer
321 views

Relative de Rham Cohomology is Homotopy Invariant

Suppose $ f:N\rightarrow M$ is a smooth map between two manifolds. Relative de Rham cohomology is defined through the complex $ \Omega^{q}(f)=\Omega^{q}(M)\oplus\Omega^{q-1}(N)$ with $d(\omega,\theta)=...
15
votes
2answers
968 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
4
votes
0answers
186 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
1
vote
0answers
30 views

existence of a closed curve and sequence that…

a) Prove that there is a curve $\alpha$ and sequence $\{x_n\}_{n \geq 0}$ in $\Bbb R^2$ such that $wind_{xn}(\alpha)=n$ for all $n \geq 0$. ($wind_{x_n}(\alpha)$ refers to the winding number of $\...
0
votes
2answers
76 views

Is it true that $X\simeq S^2\vee S^2$?

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for every $x$ in the equator $S^1$. Is it true that $X\simeq S^2\vee S^2$, that is, $X$ is homeomorphic to $S^2\vee S^2$?
1
vote
0answers
16 views

Fibrewise normal, but not functionally normal space

In general topology, if a space is normal, then exist a continuous function which separates two closed sets. This is because on a normal space, you can "put" an open set and it's closure between an ...
1
vote
0answers
80 views

A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
1
vote
1answer
64 views

Determining if certian properties of a topological space pass to its image under a quotient map.

A property $P$ of topological spaces is said to "pass to quotients" if whenever $p : X \rightarrow Y$ is a quotient map and $X$ has property $P$ then $Y$ has property $P$. For the following properties ...
4
votes
1answer
172 views

Question about Hatcher's book CW complex

I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood $N_{\...
7
votes
1answer
335 views

Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
1
vote
1answer
80 views
1
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1answer
42 views

Proving the left lifting property for a map

I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration $q:X \rightarrow Y$, where $q$ is a weak equivalence. I don't know how to draw a square ...
3
votes
0answers
104 views

Winding number and homotopy

Given two maps $f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them. Well, we know that we can write them as $f(\exp(2 \pi i t))...
1
vote
1answer
87 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
1
vote
1answer
40 views

In the definition of $n$-equivalence, what is the motivation for only requiring surjectivity on the $n$th dimension.

An $n$ equivalence $f\colon X \to Y$ such that the induced map on the homotopy group $f_* \colon\pi_m(X) \to \pi_m(Y)$ is an isomorphism for $m<n$ and an epimorphism for $m=n$. What's the ...