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

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86 views

Is $BG =EG / G$ a CW complex?

am currently working through J.Rosenbergs construction of classifying Spaces for the +-construction of higher K-Theory. He defines a CW-complex EG as the direct limit of the inductively defined Spaces ...
4
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1answer
83 views

Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
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1answer
36 views

Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.

I am trying to do this question taken from Hatchers algebraic topology and I am struggling to understand the notation and the concepts. As far as I know $\pi_1(X,x_0)$ is the set of end point ...
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1answer
56 views

$H^{n}(M)$ where $M$ is compact, orientable and connected manifold

I need to show that if $M$ is compact, orientable and connected manifold of dimension $n$, then $H^{n}(M) = \mathbb{R}$. I saw that a possible proof is to take an atlas for $M$, with $U_{\alpha}$, ...
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1answer
47 views

Computation of 2nd homology using Hopf's formula

Let $G$ be a group and $G'$ be a group obtained from $G$ by adding a one generator $x$ and relations $gx=xg$. That is, $G'=G\times\mathbb{Z}$ and $H_2(G')=H_2(G)$. Problem. Prove $H_2(G')=H_2(G)$ ...
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1answer
63 views

Hatcher 3.1.4 What happens if one defines homology groups of the chain complex?

What happens if one defines homology groups $h_n(X,G)$ of the chain complex $\cdots \rightarrow Hom(G,C_n(X)) \rightarrow Hom(G,C_{n-1}(X))\rightarrow \cdots $ ? More specifically, what are the ...
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2answers
62 views

Is the Riemann surface for the square root simply connected?

I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...
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1answer
65 views

Why can't this triangulate $\mathbb{RP}^2$?

I understand that an actual minimal triangulation of $\mathbb{RP}^2$ has at least 10 2-simplices, but I don't understand why. Without appealing to the computation of the homology groups of ...
4
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1answer
53 views

Fibered product of Hopf Fibrations

I am wondering: what is the vector bundle associated to the Hopf-fibration $S^{3}\to S^{2}$? What is the fibered product of two Hopf-fibrations? Thanks.
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1answer
38 views

Mapping cylinder cofibration

Let $f:X\to Y$ be a continuous map, and let $M_f = (X\times I) \sqcup Y)/(x,0)\sim f(x)$ be its mapping cylinder. Then the inclusion $X\to M_f$ is a cofibration. My attempt: Using the following ...
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1answer
76 views

Hatcher's Algebraic Topology, Example 1.35

Hatcher considers the mapping cylinder A from $S^{1}$ to $S^{1}$ under the function $z \rightarrow z^m$. He claims without explanation that the universal cover of A is homeomorphic to a product $C_m ...
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2answers
51 views

Compact universal covering spaces

Let $X$ be a topological compact space admitting a universal covering $C$. When is $C$ again compact? Thanks.
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1answer
55 views

hatcher's book about the wedge sum and a deformation retract

I was reading Hatcher's book ,and I can't really understand how to get the wedge sum ,is it just that i combine my spaces with a single point ? and for example he says that the complement R^3 -A ...
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3answers
258 views

Lie group. How is manifold defined?

If I have the Lie group $SL(2,\mathbb{R})$. Then how is the manifold structure on this algebraic group defined, could anybody explain this to me? I mean this is the group of matrices that determinant ...
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1answer
52 views

Power of complex number is the degree of map

I was given an off-the-cuff question in my topology class, the image below is copied exactly from the whiteboard: Here are the explanation: (1) The vertical map $e = \mathbb R \to S^1$ is the ...
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1answer
61 views

Nondegenerate points - Inclusion is a cofibration

Let $X$ be a locally Euclidean metric space and $x\in X$. Then the inclusion $x\to X$ is a cofibration. My attempt: I'm using the following result from Bredon: Let $U\subset X$ be a nbhd of $x$ ...
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1answer
24 views

Inclusion is cofibration iff $(A\times I) \cup (X\times \{0\})$ is a retract of $X\times I$

Let $A\subset X$ be a subspace. Then the inclusion $i:A\to X$ is a cofibration iff $(A\times I) \cup (X\times \{0\})$ is a retract of $X\times I$. I've proved the "$\implies$" direction. ...
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0answers
81 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
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1answer
37 views

Why is the torus not a boundary of a 3-chain?

I'm learning about homology right now and the author simply states that the torus $T^2$ does not have a boundary (I understand this) and also is not a boundary of a 3-chain. This is not at all obvious ...
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3answers
73 views

A quick/geometric reason why Hatcher's reparameterizations work in the proof of Proposition 1.3?

During the proof of Proposition 1.3 in Hatcher, Algebraic Topology, (the result that $\pi_1(X,x_0)$ is a group with respect to the product $[f][g]=[f\cdot g]$) he uses some reparamaterization tricks ...
2
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1answer
41 views

The spectral sequence of the path fibration of $S^2$

Bott and Tu use the fibration $\Omega S^2 \to PS^2 \to S^2$ to compute the cohomology of $\Omega S^2$. They do this by looking at the (Serre?) spectral sequence. Then $E_2^{p,q} = H^p(S^2,H^q(\Omega ...
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1answer
23 views

Does the elementary knot move really preserve the orientation?

So in this picture, the first diagram changed to the third diagram by the elementary knot moves, but the orientations of the first and the third are different. I wonder if in $R^3$ the moves don't ...
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1answer
40 views

What exactly is meant by “an integer basis of the $\mathbb{Z}$-module $H^*(M)$”?

I thought I understood the concept of a cohomology ring, but am confused by the following statement found in a textbook. Context: M is a symplectic manifold of dimension $2n$. "Let us choose an ...
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1answer
44 views

Is the Pontrjagin-Thom-map null-homotopic if the normal bundle $N_i$ is trivial?

Let $i\colon X\to Y$ be an embedding of two smooth and compact manifolds (without boundary) and let $N_iX$ be the normal bundle of this embedding. A Pontrjagin-Thom construction is a map $$ c_i\colon ...
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1answer
54 views

CW complex structure of sphere with identified poles

I am trying to figure out the $CW$ complex structure on a sphere with the north and south pole identified. I've been told the structure is the following Start with a $0$-cell $x$. Attach an ...
0
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1answer
33 views

Simplicial approximation

One of the definition of simplicial approximation says that: a simplicial map $h:|K|\rightarrow|L|$ is a simplicial approximation of a continuous map $f:|K|\rightarrow|L|$ if and only if $$\forall ...
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0answers
26 views

universal cover homotopy equivalent if the base space homotopy equivalent

I am working on Hatcher's algebraic topology book and I got stuck in problem 8 in section 1.3. It says if $\hat{X}$ and $\hat{Y}$ are simply-connected covering space of the path connected, locally ...
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1answer
300 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
4
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0answers
15 views

Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
2
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1answer
64 views

CW complex structure on standard sphere identifying the south pole and north pole

I need to find CW complex structure in the sphere by identifying the north and south pole. First of all I tried to visualize what it looks like and if I am not wrong gluing the poles together will ...
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0answers
16 views

Covering space BSO_n-> BO_n

Can somebody tell me where I can find information about the covering space $BSO_n \to BO_n$. Thanks in advance!
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0answers
22 views

What tools in algebraic topology help me capture the connectivity structure of a weighted graph as a REAL number?

All - This is a follow-up to a previous question about cohomology. I am researching a problem and, as with so much problem-solving, this has led me into parts of math well beyond where I went in ...
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2answers
49 views

Homotopy equivalence between $\mathbb R^3\setminus\text{$x$-axis}$ and $S^1$ [closed]

Let $X=\mathbb R^3\setminus \{(x,y,z) \in \mathbb R^3 \mid y=z=0\}$. Show that $X$ is homotopy equivalent to $S^1$. How to do it?
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0answers
42 views

Inductively Constructing Chain Homotopies

Let $f,g:C_\bullet \rightarrow D_\bullet$ be chain maps. Let $T$ be a generator (possibly not the only one) of $C_n$ and assume $H_n(C)=0$. I'm trying to prove that for every $T$ there's a solution ...
2
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1answer
34 views

Showing that identity and g are not homotopic (without Homology)

Question: Are the identity mapping on $S^1$ and the reflection about the $x$-axe homotopic? This is a question which I already know the answer. The objective is to find better answers and suggestions ...
3
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1answer
51 views

Aspherical but not contractible

Let $X$ be the topologist's sine curve (i.e. $\left\lbrace (x,y): y=\sin\left(\frac{1}{x}\right),x\in ]0,1]\right\rbrace\cup \lbrace (0,y): y\in [-1,1]\rbrace$) with an arc joining $(0,0)$ and ...
3
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1answer
91 views

How can I understand cohomology theories in the context of basic homology theory?

Please pardon the ignorance in advance -- I'm doing research, trying to solve a specific problem, so naturally I'm led down paths in mathematics I never had the opportunity to study in depth. I ...
2
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1answer
65 views

ordinary cohomology from equvariant cohomology

Is it possible that the ordinary cohomology of a space can be obtained from its equivariant cohomology? action is algebraic torus action and space is nonsingular complete complex algebraic variety ...
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1answer
25 views

Jordan regular Representative of $H_1(\Omega)$ with coefficients $\mathbb Z/ 2 \mathbb Z$

Consider the first homology group $H_1(\Omega)$ with coefficients in $\mathbb Z/2\mathbb Z$ for a bounded, open subset $\Omega\subset \mathbb C$. Then I should be able to find a representative path, ...
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2answers
63 views

Explicit construction of Eilenberg-Maclane spaces with n=1

Is there any examples of explicit construction of Eilenberg-Maclane spaces $K(G,1)$ for concrete groups except for G=$\mathbb Z$ and $\mathbb Z_n$? I know about general simplicial bar construction, ...
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1answer
26 views

Enumerating fiber bundles with fiber and base first Eilenberg-Maclane spaces

How to enumerate fiber bundles (maybe, only as spaces, not bundles) with fiber $K(A, 1)$ and base $K(B, 1)$? It seems to be connected with enumerating short exact sequences of form $0 \to A \to ... ...
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1answer
52 views

Is it possible to continuously choose one-dimensional subspace in each k-dimensional subspace?

Does there exist a continuous map from Grassmann manifold to projective space $Gr^n(V) \to \mathbb P(V)$, such that image of every n-dimensional subspace lies (1-dimensional subspace) in this ...
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0answers
40 views

A basis for the profinite topology

Let $G$ be a group, and let $$C =\{H\leq G :\ [G:H]<\infty \}$$ $$B_R =\{Hx\leq G :\ H\in C ,x\in G\}$$ $$B_N =\{Hx\leq G :\ H\in C ,H\ is\ normal\ in\ G,x\in G\}$$ I want to show that $B_R$ ...
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63 views

a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

I've been wondering about such problems. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$. But does there exist a space $X\subset\mathbb R^3$ (maybe even ...
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1answer
60 views

A question about the Möbius Strip and the Projective Plane

I know that both the Möbius Strip and the Projective Plane are both 2-manifolds. I try to prove that they are locally homeomorphic to $\mathbb{R}^2$ and Hausdorff. It seems easy to see that the ...
3
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1answer
47 views

Subgroups of free products of cyclic groups

Consider the free product $\mathbb{Z}_{3} \star \mathbb{Z}_{3}$. How would one determine the number of subgroups of this product up to isomorphism? It is routine for the case of the product ...
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32 views

Find the fixed points of the action

Let $T$ be the set of all invertible diagonal matrices of determinant one. How can I find out the fixed points and $1$-dimensional orbits of $T \times T$ action on $\mathbb{CP}^3$ given by $(A,B)\cdot ...
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2answers
38 views

Universal covering space of X x classifying space of \pi_1(X)

I am trying to learn about classifying spaces for a Lie group $G$. The question I have is the following: Suppose $X$ is a manifold and $G=\pi_1(X)$ is its fundamental group, is it true that ...
3
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1answer
48 views

Weak equivalence testable on invariant open covers?

Let $f\colon X\rightarrow X'$ be a continuous map between two spaces $X,X'$, which might be arbitrary wild, especially I don't want to work in any convenient category of topological spaces. Let ...
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0answers
14 views

Is the projection from the cyclic nerve to the simplicial nerve a cyclic map?

I’m trying to understand Loday’s proof of Theorem 7.3.11 in “Cyclic Homology” (2nd ed 1998). I have a problem right at the beginning where it says: The projection map $\text{proj}:\Gamma_\bullet G ...