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

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2answers
42 views

Given the connectivity of $A$ and $A\wedge B$, what can be said about the connectivity of $B$?

Let $A$ and $B$ be well pointed CW-complexes. I guess an answer to the question in the title is, that nothing can be said about the connectivity of $B$. Therefore, I ask for a counter example: ...
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2answers
94 views

Example of $H^n(X,R)$ not equal to $Hom(H_n(X,R),R)$

The universal coefficient theorem shows that under suitable assumptions, the cohomology groups with coefficients in $R$ are simply the morphisms between the homology groups and $R$. In general, ...
3
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2answers
76 views

When the Induced Homomorphism on the $n$-th Cohomology is an Isomorphism

I am trying to show that when you're given a continuous map $f:M\rightarrow N$ between compact orientable $n$-dimensional manifolds and $f^*:H^n(N)\rightarrow H^n(M)$ is an isomorphism, then ...
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1answer
33 views

Fixed point property of “3-star”

Let $X = (I_1\sqcup I_2 \sqcup I_3)/(0_1 \sim 0_2\sim0_3),(I_i=[0,1]_i)$. I spend much time to trying to prove that any continuous map $X\to X$ have fixed point, but with no results..
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1answer
29 views

Cellular homology boundary maps of a closed orientable surface of genus g

When computing homology of a closed orientable surface of genus g we get the following chain complex in cellular homology: $0 \rightarrow \mathbb{Z} \xrightarrow{d_2} \mathbb{Z}^{2g} ...
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1answer
74 views

dual basis of cohomology algebra

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients. Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$. Let the dual basis be ...
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4answers
85 views

The Fundamental group of Klein Bottle

My question is if $$\pi_{1}(KB)\cong\frac{\mathbb{Z}(a)}{\langle a^{2} \rangle}*\frac{\mathbb{Z}(b)}{\langle b^{2}\rangle}\cong\frac{\mathbb{Z}(a)*\mathbb{Z}(b)}{\langle a^{2}b^{2} \rangle}$$ and ...
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0answers
43 views

Curvature of a principal bundle and the exterior covariant derivative

Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a connection on $P$, where $\mathfrak{g}$ is the Lie algebra of $G$. Associated to every ...
4
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0answers
91 views

Yoneda-type lemma for compositions on the hom-functor

The Yoneda lemma basically rephrases a rigidity property of natural transformations out of a covariant hom functor: A natural transformation $\psi : \mathsf{Hom}(Z,-)\Rightarrow F$ is determined by ...
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1answer
39 views

On embedding a sort of $CW$ complexes to a Euclidean space.

I'd like to know if a finite dimensional, locally finite, $CW$ complex with countable cells can always be embedded to a Euclidean space. All I know is that it holds in the case $\dim=1$.
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1answer
53 views

Isometry of two Euclidean structures on the same vector bundle

I'm reading Milnor & Stasheff's Characteristic Classes, and I was struck by the following problem, which they call the Isometry Theorem: "Let $\mu$ and $\mu'$ be two different Euclidean metrics ...
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3answers
53 views

How is the following a CW complex

My professor today draw on the board a sphere and attached to half a circle of the sphere half of the boundary of a disk so the shape looked like you glue a curvy half disk to a sphere. He then said ...
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1answer
32 views

Why are cohomotopy groups defined only up to dimension $2m-2$ and not $2m-1$?

The addition in $\pi^m(X)=[X,S^m]$ is defined as follows. Choose representants $\alpha, \beta$ of $[\alpha], [\beta]\in \pi^m(S)$ and let $(\alpha, \beta): X\to S^m\times S^m$. If $\mathrm{dim} ...
2
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0answers
74 views

Manifolds and CW-complexes [duplicate]

This is a very naive question. Every manifold (assumed to be paracompact) is a CW-complex? Thanks.
1
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0answers
32 views

Connection between Chladni Plates and Algebraic Topology?

Does anybody know of a connection between Chladni Plates and Algebraic Topology? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
3
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2answers
79 views

Vector bundle of dimension $\leqslant n$ on $n$-connected space is trivial

I wonder whether any vector bundle of dimension $\leqslant n$ on an $n$-connected CW-complex is trivial? It seems that, the complex can be given cellular structure with exactly one 0-dimensional ...
4
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2answers
113 views

Showing that two spaces are homeomorphic

I was trying to show that a torus is homeomorphic to $S^1 \times S^1$ , I tried to work with the fundamental group of both, which are equal, but that doesn't imply they're homeomorphic, (at least i ...
2
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0answers
35 views

Poincare-Lefschetz duality, universal coefficients, and middle cohomology

Sorry if the question is too simple, algebraic topology is not my strong suit. Let $(M,\partial M)$ be a $2n$-dimensional manifold with boundary, with one-dimensional middle cohomology. By ...
3
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1answer
85 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?
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0answers
38 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
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1answer
44 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?
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1answer
84 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 ...
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0answers
33 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 ...
2
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0answers
63 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
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1answer
85 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
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1answer
43 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
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1answer
69 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 ...
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0answers
34 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
70 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?
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0answers
22 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
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2answers
220 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 ...
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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
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1answer
64 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.
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0answers
65 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 ...
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1answer
62 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 ...
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1answer
43 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 ...
2
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1answer
24 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 ...
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1answer
39 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
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1answer
92 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
86 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
37 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
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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 $U_{\alpha}$, ...
5
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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
65 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
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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
votes
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.
2
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1answer
40 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
54 views

Compact universal covering spaces

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