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

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2
votes
1answer
61 views

Find $f$ and $g$ homotopic s.t. induce different homomorphisms

Let $X$ and $Y$ be two topological spaces. Are there continuous functions $f,g:X\to Y$ satisfying the following conditions? $f(a)=g(a)=b$ for some $a\in X$, $f$ and $g$ are homotopic, and the ...
8
votes
1answer
113 views

Topological invariants by integrals

Some topological invariants that can be found e.g. in knot theory can be represented as integrals (Example: Integral for computing the Gauss linking number). Another example is the complex plane with ...
11
votes
2answers
341 views

Does the ham sandwich theorem hold for dividing objects into thirds?

The ham sandwich theorem states that given $n$ measurable "objects" in $n$-dimensional space, it is possible to divide all of them in half (with respect to their measure) with a single ...
3
votes
2answers
118 views

How to construct $K(\mathbb{Z}/5\mathbb{Z},1)$?

From the Wikipedia article on Eilenberg-MacLane spaces: A $K(G, n)$ can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group ...
1
vote
0answers
41 views

The +-construction on a homology n-sphere

I am working on Weibel's K-Book and when defining higher K-Theory for a Ring via $BGL(R)^+$, I have encountered a question concerning a homology n-sphere. The statement I want to show is the ...
2
votes
1answer
51 views

Crossed module structure on $\pi_1$-level of any map $f: X\to Y$

For cofibration $f:A\to X$ we have crossed module $\pi_2(X,A)\to\pi_1(A)$. On other hand, we can change map $f$ to fibration and consider crossed module $\pi_1(E)\to\pi_1(P_f)$, where $P_f$ is ...
1
vote
0answers
96 views

Is a variety a CW-complex?

How to establish that any differentiable manifold and any complex algebraic variety is a CW-complex ? Thank you in advance for your help.
0
votes
0answers
67 views

Connections between probability theory and algebraic topology?

Are there any substantial connections between probability theory and algebraic topology? In particular, are there any current research areas in algebraic topology that involve the use of probabilistic ...
4
votes
2answers
312 views

Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$

Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$. I can image how to do by the graph,as think of $RP^2$ as the unit disk with opposite boundary points ...
0
votes
0answers
23 views

Metric, torsion free connections on principal bundles

Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ...
1
vote
1answer
131 views

Integration of forms on non-simply connected manifolds

What I know is that closed forms are not exact on non-simply connected manifolds, so for instance, if $E$ is a closed form, then $dE = 0$ but $\int_\gamma E \neq 0$, where $\gamma$ is a ...
1
vote
1answer
62 views

Vector bundle over a compact, Hausdorff space is a summand of a trivial bundle.

I am trying understand the proof of the following (proposition 1.4 in Hatcher's book on Vector Bundle). For every vector bundle $E\overset{p}{\to} B$, with $B$ compact Hausdorff, there exists a ...
2
votes
1answer
68 views

A quotient map $X\to X/A$ that is not a Serre fibration

What is an example of a CW-pair $(X,A)$ such that the quotient map $X\to X/A$, i.e. the map obtained from the pushout \begin{eqnarray} A &\to& X\\ \downarrow &&\downarrow\\ * ...
0
votes
1answer
46 views

Hatcher $3.1.12$ Show that $H^k(X,X^n;G)=0$ for $k \leq n$

In Hatcher it is written in a theorem that by Universal Coefficient Theorem we get $H^k(X,X^n;G)=0$ where $X,X^n$ are CW complexes. But to use UCT, we have to show $H_k(X,X^n;G)=0$ $k \leq n$. ...
0
votes
2answers
48 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: ...
1
vote
2answers
106 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
votes
2answers
142 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 ...
1
vote
1answer
42 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..
0
votes
1answer
66 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} ...
1
vote
1answer
104 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 ...
0
votes
4answers
139 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 ...
3
votes
0answers
150 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 ...
6
votes
1answer
162 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 ...
0
votes
1answer
57 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 $\dim=1$.
1
vote
1answer
77 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 ...
0
votes
3answers
88 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 ...
1
vote
1answer
35 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
votes
0answers
81 views

Manifolds and CW-complexes [duplicate]

This is a very naive question. Every manifold (assumed to be paracompact) is a CW-complex? Thanks.
1
vote
0answers
36 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
votes
2answers
92 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
votes
2answers
132 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
votes
0answers
88 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
votes
1answer
102 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?
1
vote
0answers
48 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
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?
3
votes
1answer
149 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 ...
0
votes
0answers
45 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
votes
0answers
79 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, ...
9
votes
1answer
249 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
57 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
97 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
46 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
74 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
44 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 ...
8
votes
2answers
335 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
24 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
91 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.
6
votes
0answers
131 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
76 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 ...
1
vote
1answer
117 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 ...