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

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-4
votes
1answer
29 views

Examples of a infinite dimensional simplicial. [on hold]

I want see an example of infinite dimensional simplicial, diferent to the examples built using ergodic theory.
0
votes
0answers
42 views

What is on the cover of Hatcher's Algebraic Topology book?

What is on the cover of the book? Is it the Hopf fibration?
0
votes
0answers
21 views

Cohomology ring of classifying space

I am looking for $H^*(BZ/2p , Z/2p)$ where $p$ is odd prime.We can calculate cohomology groups by using gysin exact sequence and universal coefficient theorem.But I am unable to calculate the ring ...
0
votes
1answer
35 views

A topological question of division

Is there a relation between $K(\Bbb Z[\frac{1}2],1)$,$K(\Bbb Z,1)$? Why is $K(\Bbb Z[\frac{1}2],1)$ like $K(\Bbb Z,1)$ 'divided' by 2?
3
votes
1answer
56 views

Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
-2
votes
0answers
29 views

Inmersion of a $\beta\in\pi_{1} (X,x_{0})$ [on hold]

Let $X$ be a topological space, for any integer $n$, let $X_{n}$ be a arcwise-connected subspace which have the base point $x_{0}\in X.$ And suppose that $X_{n}\subset X_{n+1}$ for any n, ...
0
votes
0answers
29 views

How do you show that the pinched torus is a pseudomanifold?

How do you show that the pinched torus is a pseudomanifold? This is a pinched torus: A topological space $X$ endowed with a triangulation $K$ is an $n$-dimensional pseudomanifold if the ...
1
vote
0answers
53 views

A homotopy equivalence between two sets

I was trying to prove that the set consisting of the union of the circles $\{\langle x,y\rangle\mid(x-10)^2 +y^2 = 1\}$, $\{\langle x,y\rangle\mid(x+10)^2 +y^2 = 1\}$ and line segment $\{\langle x, ...
7
votes
1answer
70 views

$S^2=SO(3)/SO(2)$. Does this mean that $S^2 = SU(2)/U(1) $?

$S^2=SO(3)/SO(2)$. Does this mean that $S^2 = SU(2)/U(1) $ since $SO(3) \approx SU(2)$ and $SO(2) \approx U(1)$? Is there some more generic rule on how to relate $S^{n-1} = SO(n)/SO(n-1)$ to the ...
1
vote
1answer
35 views

Homotopy and Semidirect Product

I know there is a relation in homotopy theory which is $\pi(G\times H) = \pi(G)\times\pi(H)$. However, does this relation still hold for $\pi_0$ which may not be a group? Moreover, is there such a ...
2
votes
1answer
50 views

When a homotopy equivalence of the closed unit ball in the Euclidean n- dimensional space is injective?

I have a compact connected metric space $X$ of dimension $n$ which is homotopically equivalent to the closed unit ball $D^n$ in the n-dimensional Euclidean space. I am wondering if there is an ...
1
vote
0answers
25 views

$\Delta$-complex structure on $S^{2n-1}$

Does there exist a $\Delta$-complex structure on $S^{2n-1}$ by identifying pairs of faces of $\Delta^{2n-1}$ with only one $(2n-1)$-simplex? (where $\Delta^n=\{(x_0,x_1,...,x_n)\in ...
3
votes
2answers
62 views

If $A$, $B$ are path connected and $A \cup B$ is simply connected, $A \cap B$ is path connected

The only proof I know involves the Mayer - Vietoris sequence. Is there an elementary proof?
0
votes
0answers
15 views

Skein relationship and Alexander polynomial

Given is the three link diagrams of Conway $L_{0},L_{+},L_{-}$ and the corresponding Seifert matrices $M_{0},M_{+},M_{-}$. Prove that ...
5
votes
2answers
189 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
0
votes
0answers
17 views

rational cohomology of finite dimensional grassmannian

Let $G$ denote the grassmannian. It is known that the cohomology ring $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Q})=\mathbb{Q}[p_1,p_2,\cdots,p_{[n/2]}]. $$ What is $$ ...
0
votes
0answers
16 views

Universal G-bundle

I want to study the cohomology of the bundle $BSO_n \times BSO_m \to B[O_n \times O_m]^{+} $, where $[O_n \times O_m]^{+} = (O_n \times O_m) \cap SO_{n+m}$. I know that for studying such cohomology I ...
2
votes
1answer
51 views

Associativity of concatenation of closed curves from $I$ to some topological spaces $X$

I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$. I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
5
votes
0answers
53 views

An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
0
votes
1answer
57 views

allen hatcher page 46 ex 1.23

I've recently asked a question about Hatcher's explanation of the deformation retraction of $R^3-A$, where $A$ a circle, to the wedge sum of $S^1$ & $S^2$ (page 46, ex 1.23). I didn't get an ...
0
votes
1answer
33 views

Definition of topological group acting on a topological space

The definition of a topological group $G$ acting on a topological space $X$ is there exists a continuous map from $G\times X \rightarrow X$ such that $e_G.x=x$ for all $x\in X$ and ...
1
vote
1answer
20 views

Is restriction of a chart is a chart necessarey in the Definition of Fibre Bundle

This is the definition of Fibre Bundle from the notes James F Davis and Paul Kirk: I think the condition 3 is superfluous. Because if you have a chart over $U$ $\phi : U \times F \rightarrow ...
1
vote
1answer
33 views

Show any homomorphism from $\pi_1(S^1)$ to itself is an induced homomorphism of some $f:S^1\rightarrow S^1$

Q/ Show that any homomorphism $\phi:\pi_1(S^1)\rightarrow \pi_1(S^1)$ can be realised as the induced homomorphism $f^{*}$ of a map $f:S^1\rightarrow S^1$. A/ $f$ induces $\phi$ if ...
1
vote
1answer
59 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 ...
2
votes
1answer
39 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), ...
1
vote
1answer
34 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 ...
1
vote
1answer
51 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 $U_{\alpha}$, ...
5
votes
1answer
42 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 $H_2(G')=H_2(G)$ ...
1
vote
1answer
56 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 ...
0
votes
2answers
49 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 ...
2
votes
1answer
60 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
votes
1answer
45 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.
2
votes
1answer
29 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 ...
2
votes
1answer
72 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 ...
1
vote
2answers
46 views

Compact universal covering spaces

Let $X$ be a topological compact space admitting a universal covering $C$. When is $C$ again compact? Thanks.
1
vote
1answer
47 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 ...
6
votes
3answers
243 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 ...
1
vote
1answer
51 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 ...
2
votes
1answer
54 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 $x$ ...
3
votes
1answer
22 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. ...
4
votes
0answers
51 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 ...
1
vote
1answer
32 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 ...
1
vote
3answers
70 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
votes
1answer
38 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 ...
0
votes
1answer
21 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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
41 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 ...
0
votes
1answer
60 views

a question about undergraduate differential geometry, how to prove k=$|k_{n}k_{N}|$?

Can someone help me to prove:If $C=\alpha(I)$ is a line of curvature, and k is its curvature at p,then $$k=|k_{n}k_{N}|$$, where $k_{n}$ is the normal curvature at p along the tangent line of C and ...
1
vote
0answers
35 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
votes
1answer
31 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 ...