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

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1answer
98 views

Betti number and the homology class - what determines the coefficient $Q$?

From Wikipedia: For a non-negative integer $k$, the $k$th Betti number $b_k(X)$ of the space $X$ is defined as the rank (number of generators) of the abelian group $H_k(X)$, the $k$th homology group ...
2
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1answer
89 views

An example of why this homomorphism of homotopy groups is not injective.

For a given path-connected space $X$, I recently learned that one could construct a CW complex $X_{1}$ by considering each generator $j: S^{q} \rightarrow X$ of $\pi_{q}(X)$ and setting ...
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1answer
46 views

How to calculate homotopic groups : $\pi_n(Z)$ and $\pi_n(S^0)$ .

While doing an exercise, I need to show that $Z$ and $S^0$ are not homotopically equivalent. To do so, I'd like to show that $\pi_n(Z) \neq \pi_n(S^0)$ for some $n$. But I can't figure out if to ...
2
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1answer
56 views

Reference for secondary cohomology operations

I am learning some homotopy theory and am currently reading Mosher and Tangora. I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very ...
3
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1answer
207 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
3
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1answer
127 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
6
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1answer
187 views

Comparing notions of degree of vector bundle

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand ...
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2answers
157 views

The Zariski density for two given sets.

Let $A$ and $B$ be two subsets of $\mathbb{C}^n$: $ A = \mathbb{Z}^n$, and $B=\{ (z_1,z_2, \dots , z_n) \in A \text{ such that } z_1>z_2>\cdots> z_n\}$. My questions: Are these two subsets ...
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2answers
159 views

cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
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3answers
212 views

The fundamental group of the projective plane minus 2 points?

I'm trying to compute the fundamental group of $\mathbb{RP}^2$ minus 2 points. I'm using the presentation $\langle a\mid a^2\rangle$. Meaning that I'm taking the disk and identifying the two sides. ...
6
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1answer
120 views

Is this morphism of spectra zero in the stable homotopy category?

Let $f\colon A\to B$ a morphism of spectra and suppose that both spectra $A$ and $B$ have only one non-zero stable homotopy group $\pi_n$, more precisely $$ \pi_k(A)=0=\pi_k(B) $$ for $k\neq n$. ...
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2answers
117 views

K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
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1answer
44 views

null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity

I am reading a book suggesting that the following is true: Let $X$ be a simply connected (maybe finite) simplicial complex. Then there exists a continous map $$g : X \to X,$$ homotopic to the ...
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1answer
53 views

Relative homology groups

I have to compute the homology groups $H_{n}(X,A)$ when $X$ is $S^{2}$ or $S^{1}\times S^{1}$ and $A$ is a finite set of points in $X$. So, I write the exact long sequence : $...\rightarrow ...
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0answers
25 views

Basic question about abelianization of Homotopy Groups and Homology [duplicate]

When precisely, is the homology group: $$H_n(T)$$ of a topological space, $T$, isomorphic to the abelianiation of the corresponding homotopy group $\pi_n$? Does this only occur when $n=1$, or is ...
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1answer
114 views

The relative homology group $H_{1}(\mathbb{R},\mathbb{Q})$

This problem is an extract of Hatcher's book. Show that for the subspace $\mathbb{Q}\subset \mathbb{R}$, the relative homology group $H_{1}(\mathbb{R},\mathbb{Q})$ is free abelian and find a ...
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1answer
61 views

Rotman, Algebraic Topology, Lemma $4.22$

Lemma 4.22. Let $X$ be a space and, for $i=0,1$, let $\lambda _i:X\rightarrow X\times I$ be defined by $x\mapsto (x,i)$. If $H_n (\lambda _0)=H_n(\lambda _1)$, then $H_n(f)=H_n(g)$ whenever ...
4
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1answer
96 views

Moore space, induced map in homology

Let $A$ be a finitely generated abelian group and $n$ a positive integer. I have built a connected space $M(A,n)$ such that all its reduced homology groups are zero but the i-th reduced homology group ...
2
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0answers
104 views

mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
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0answers
56 views

Correspondence between first homology group and deck transformations.

Let $M$ be a connected topological manifold with universal covering $\pi: \widetilde{M} \rightarrow M$ and let $p \in \widetilde{M}$ be a point. Let $\alpha,\beta : \widetilde{M} \rightarrow ...
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1answer
57 views

Homotopy Invariance: Cone Construction and Prisms Operators

I'm looking at different approaches to proving the homotopy invariance of homology. Rotman and Dieck both mention "the cone construction", but hatcher only introduces the prism operators and does not ...
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0answers
40 views

Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
3
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1answer
68 views

Zero homology and path-component

I try to show the two next things : $H_{0}(X,A)=0$ iff $A$ meets each path-component of $X$ and $H_{1}(X,A)=0$ iff $H_{1}(A)\rightarrow H_{1}(X)$ is surjective and each path-component of $X$ ...
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0answers
61 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
2
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1answer
43 views

Why a convergent succesion does not have the same homotopy type of a CW-Complex?

The question is pretty much in the title; If my space is $\{1/n\}_{n\in \mathbb{N}} \cup \{0\} $ why it isn't homotopically equivalent to a CW-Complex?
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0answers
45 views

Short exactness sequence of chain complexes (singular homology)

I want to prove this claim http://planetmath.org/longexactsequenceofhomologygroups that $$0\to C_k(A,B)\to C_k(X,B)\to C_k(X,A)\to 0$$ with the maps $C_k(i):C_k(A,B)\to C_k(X,B)\; [\sigma]\mapsto ...
0
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2answers
125 views

Retract and homology

I have this problem in Hatcher's book : Show that if $A$ is a retract of $X$ then the map $H_{n}(A)\rightarrow H_{n}(X)$ induced by the inclusion $A\subset X$ is injective. I think I have ...
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2answers
73 views

Excision theorem

We consider $f:S^{n}\rightarrow S^{n}$ a continuous function. Then we consider $y\in S^{n}$ such as $f^{-1}(y)=\{x_{1},..., x_{p}\}$, $U_{1},..., U_{m}$ $x_{i}$-neighbourhoods, and $V$ y-neighbourhood ...
1
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1answer
52 views

$\mathbb RP^n$ as CW-complex

In hatcher (p.6) it says that $\mathbb RP^n$ can be obtained from $\mathbb RP^{n-1}$ by attaching an $n$-cell via the quotient map $S^{n-1} \to \mathbb RP^{n-1}$. I was wondering what this is for ...
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2answers
65 views

Manifold with $\pi_1(M)=F_n$

We may construct a 3-manifold $M_n$ with $\pi_1(M_n)\cong F_n$ (i.e. the free group on $n$ generators) as follows: consider the complement of $n$ pairs of open 3-balls in $\mathbb{R}^3$. For each ...
0
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1answer
68 views

Show that $\mathbb{R} P^3$ is not homotopy equivalent to $\mathbb{R} P^2 \vee S^3$.

I'm studying for an oral qualifying exam in algebraic topology, going through questions in various tests published on the interwebs. Here's a rather straightforward question from this exam that is ...
4
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2answers
120 views

Homomorphism between homotopy groups of spheres induced by the fibration and the multiplication map of $SO(n)$

Let $n$ be a nonnegative integer and $x\in S^n$ a point in the n-sphere. Combining the map $\alpha\colon SO_{n+1}\longrightarrow S^n$ induced by matrix multiplication with $x$ and the connecting ...
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0answers
50 views

Is the question in the Munkres's topology book wrong?

At the end of cheapter $8.1$, $4)$ Given spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. $b)$ Show that if $Y$ is path connected, the set $[I,Y]$ has a ...
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2answers
129 views

Hatcher problem 1.2.12

In this problem a modified Klein bottle (say $X$) is taken in account which is seen as embedded space in $\mathbb R^3$ (giving subspace topology on the usual self intersecting figure of Klein bottle ...
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3answers
34 views

$S^{n+m+1}$ can be decomposed as the union of $S^n\times D^{m+1}$ and $D^{n+1}\times S^m$ along their boundaries.

Let $S^n$ denote the $n$-sphere and $D^n$ denote the $n$-disk (of course, $\partial D^{n+1}\cong S^n$). Then $S^n\times D^{m+1}$ and $D^{n+1}\times S^m$ both have boundary $S^n\times S^m$. The ...
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1answer
128 views

Proving a nowhere vanishing vector field on 2D manifold implies $TU\cong M\times S^1$

So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is ...
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0answers
41 views

Non-isomorphism of topological line bundles on a Riemann surface, from first principles only

Although this question is in the same vein as my previous query, Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves, it is nonetheless ...
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1answer
51 views

Homology group of an infinite wedge product

I'm struggling with the following algebraic topology problem: I'm given a collection of (topological) spaces $(X_i)_i$, ($i\in I$ whatever) and for each space a point $x_i\in X_i$. Then $\bigvee_i ...
3
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1answer
69 views

Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
2
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2answers
103 views

Compact surface homotopy equivalent to $\mathbb{CP}^2 \setminus \{p\}$

Let $p$ be a point $\in \mathbb{CP}^2$. Is there a compact surface which is homotopy equivalent to $\mathbb{CP}^2 \setminus \{p\}$ ? I know the homology of $\mathbb{CP}^2$, but I'm not sure about the ...
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2answers
75 views

Does there always exist a lift of a path from $Y$ to $X$ if $f: X\to Y$ is a continuous surjective function?

If one has a continuous surjective function $f:X \longrightarrow Y$ and let $\gamma$ be a continuous path in $Y$, under what circumstances can one find a (possibly non-unique) lifted path $\gamma'$ in ...
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1answer
41 views

Relation between cup product and intersection number

Suppose $M$ is an oriented diff. manifold and $X$ and $Y$ are two submanifolds of codimension $m$ and $n$ in $M$. Then under some conditions one can define the intersection of $X$ and $Y$ and this is ...
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0answers
44 views

deformation retraction for a simple Hopf link

I would like to find a deformation retraction $F:X\times[0,1]\to X$ of the complement of the Hopf link which consists of two circles linked together once. Since Hatcher describes a deformation ...
3
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0answers
47 views

cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
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0answers
155 views

Need help on how to compute the fundamental group of a space.

I'm studying for an oral qualifying exam and going through various past exams I find on the interwebs, including this mock exam from the University of Bath. One of the questions seems like it should ...
0
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1answer
56 views

What does $d_{n+1}\circ{d_{n}}=0$ mean in the definition of a chain complex?

According to the Wiki article on chain complexes, a chain complex $(A_{\bullet},d_{\bullet})$ is a sequence of abelian groups or modules connected by homomorphisms such that $d_{n+1}\circ{d_{n}}=0$. ...
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1answer
90 views

tensor product of trivial line bundles

In Hatcher's book on Vector Bundles, the tensor product of two vector bundles is defined through the gluing functions. But I need an example to understand it. So I think of the simplest case, the ...
2
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0answers
136 views

Mobius band does not retract to boundary circle - specific part

(I'm trying to ask this in a way such that it isn't a duplicate question) The proofs that I've seen for the fact that there is no retraction from the Mobius band to its boundary circle usually say ...
1
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1answer
69 views

Smash product and tensor product of groups

The smash product acts like a 'tensor product' in the category of pointed spaces (i.e. when the spaces are locally compact Hausdorff smashing is associative and satisfies a tensor-hom adjunction). ...
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0answers
29 views

Right Veering Property of elements in MCG(S)

Let h be an element of MCG(S), the mapping class group of a surface S. I was going over : I was going over :Geometric Intuition for "Right-Veering" Property of $f$ in MCG(S)? Where a p.e ...