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

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1answer
43 views

Expressing generalized cohomology by ordinary cohomology

I'd like to ask for either pointing an error or confirming correctness of the following reasoning. Theorem: let $h^* \colon CW \to Ab$ be a cohomology theory, then there exist abelian groups $ ...
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1answer
16 views

Relative homotopy and composition of maps

I am trying to prove something and am stuck on the following issue : Suppose $\Psi, \Phi : I^n \to Y$ are two maps and $q:Y \to Z$ is a homotopy equivalence such that $q \Phi \cong q \Psi $rel ...
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1answer
45 views

Hatcher exercise 2.1.6 (Simplicial homology)

Compute the simplicial homology groups of the $\Delta$-complex obtained from $n+1$ $2$-simplices $\Delta_0^2,...,\Delta_n^2$ by identifying all three edges of $\Delta_0^2$ to a single edge, and for ...
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1answer
48 views

Isomorphism of CW-complexes

It is known (see for instance http://math.stackexchange.com/a/42020) that "the homotopy type of a CW complex is entirely determined by the homotopy classes of the attaching maps". What is the precise ...
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1answer
35 views

Stiefel-Whitney classes with Z-coefficients

Stiefel-Whitney classes are defined (for example, in Milnor's Characteristic classes) as elements of the cohomology groups with $\mathbb{Z}_2$-coefficients as follows. $$w_i(E)=\phi^{-1} Sq^i(u)$$ ...
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0answers
27 views

Compute a chern class $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(C)$.

This is Problem 16-D in Characteristic classes by John W. Milnor and James D. Stasheff. Problem 16-D) If the complex manifold $K^n$ is complex analytically embedded in $K^{n+1}$ with dual ...
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52 views

Question about Poincare duality and homology of a cylinder.

I am reading the paper. I have some questions about Poincare duality and homology of a cylinder. On page 9, example 2.6. Let $X = \mathbb{R} \times S^1$ be a cylinder and $Y = X/(0 \times S^1 )$, ...
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0answers
39 views

I need a good reference in topology [duplicate]

Can someone please give the title of a good topology book with exercises, preferably written by a master in the field Actually, i have basic notions like compactness, completeness, connectedness and ...
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0answers
44 views

Commutative diagram of cohomology (to show Albanese variety is a torus)

Suppose $X$ is a compact Kahler manifold of complex dimension $n$, define $H_1(X,\mathbb{Z})\to H^0(X,\Omega_X^1)^*$ by $[\alpha]\to \int_\alpha\cdot-$. We want to show the image of ...
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1answer
47 views

conditions for two automorphisms of a covering space be homotopic?

If $X\rightarrow Y$ is a covering space, and $\sigma\ne\tau\in Aut(X/Y)$. Under what assumptions on $X,Y,\sigma,\tau$ would $\sigma,\tau$ be homotopic? Edit: I'm specifically interested in the case ...
2
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1answer
50 views

union and sum induce isomorphism in homology

I am reading Hatcher's book. I have some difficulty understanding the relative cup product. In the proof it says if A and B are open in X, the inclusion $C^n(X, A\cup B;R)\rightarrow C^n(X, A+B;R)$ ...
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1answer
33 views

Computing $\pi_2$ of some 2-complex.

Let $m$ be a positive integer and $F_m$ be the free group on $m$ generators. Choose an element $\gamma\in F_m$. Then, there is a loop $\overline{\gamma}\colon S^1\to \vee_{i=1}^m S^1$ such that ...
2
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1answer
54 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 ...
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1answer
86 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 ...
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2answers
280 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
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2answers
92 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 ...
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0answers
29 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
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1answer
46 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 ...
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0answers
75 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.
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0answers
33 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 ...
3
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3answers
86 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 ...
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0answers
19 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 ...
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1answer
91 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 ...
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1answer
39 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 ...
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1answer
34 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\\ * ...
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1answer
25 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 $H_k(X,X^n;G)=0$ $k \leq n$. ...
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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
93 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, ...
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2answers
75 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
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1answer
32 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
84 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 ...
2
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0answers
42 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
89 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
31 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
33 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 ...
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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?
3
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1answer
79 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
32 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
62 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, ...