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

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0
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1answer
32 views

$(\mathbb{Z}/n\mathbb{Z})$-homology isomorphism is also a $(\mathbb{Z}/n^k\mathbb{Z})$-homology isomorphism

I'm trying to prove that if a map $f \colon X \to Y$ induces isomorphisms on singular homology with coefficients in $\mathbb{Z}/n\mathbb{Z}$, then the same is true for coefficients in $\mathbb{Z}/n^k\...
2
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1answer
73 views

Homologically trivial immersion

Are there any examples when some manifold $N$ maps in other manifold $M$ as codimension 1 submanifold, its fundamental class is zero in the homology of M, but still this map $i\colon N\to M$ induces a ...
1
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1answer
51 views

Proving Hopf degree theorem using Pontrjagin-Thom isomorphism

Does anyone know a good reference which proves Hopf degree theorem using Pontrjagin-Thom theorem, that is passing to the determination of framed bordism classes of 0-manifolds? Many thanks! Hopf ...
1
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2answers
47 views

Adjoint theorem for loop and suspension

Ref: Davis and Kirk, Lecture Notes on Algebraic Topology On pp.114 it is the adjoint theorem for the category of Hausdorff compactly generated spaces: for $X,Y,Z$ compactly generated $f(x,y) \mapsto \...
1
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0answers
46 views

Obstruction to lifting a map from the base space to the total space.

Suppose $\pi :E \to B$ is a fibration with fibre $F$ above a chosen base point. Then I am trying to solve when a map $f$ from a manifold $M$ to $B$ lift to a map $g:M \to E$. The answer given is they ...
5
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1answer
63 views

Covering space is path-connected if the action of $\pi_1$ on a (single) fiber is transitive

Let $p\colon X\to Y$ be a covering map. Suppose that $Y$ is path-connected, locally path-connected and semi-locally simply connected. Let $x,x'\in X$ be two points of $X$. $\textbf{Question:}$Is ...
2
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1answer
65 views

problem 14 of section 1.2 from Hatcher

Consider the quotient of a cube $I^3$ obtained by identifying each square face with the opposite square face via the right-handed screw motion consisting of a translation by one unit in the direction ...
5
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2answers
153 views

Why is this easy “proof” of Brouwer's Fixed Point Theorem not correct/common?

Brouwer's Fixed Point Theorem states, essentially, that any continuous function on a closed disc to itself has a fixed point. I am familiar with the proof based on the impossibility of a retraction ...
2
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2answers
79 views

If two space X and Y have isomorphic fundamental group then when can we say that this isomorphism is induced by some map f from X to Y?

If two space X and Y have isomorphic fundamental group then when can we say that this isomorphism is induced by some map f from X to Y ? and also what can we say about this question when we take ...
2
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1answer
37 views

When do the lifting properties characterize covering spaces?

A covering map $p:E \rightarrow B$ is continuous map satisfying the following: For every point $b\in B$, there exists a neighborhood $U$ of $b$, such that $p^{-1}(U)$ is a disjoint union of open ...
0
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1answer
40 views

Nested sequence of compact connected sets

Suppose that $K_1 \supset K_2 \supset K_3 \supset \dots $ is a nested sequence of compact connected subsets of $S^2$ such that $\pi_1(K_j)\simeq \mathbb{Z}$ for all $j$. Prove or provide a ...
2
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0answers
53 views

$B\subseteq A \subseteq \mathbb{R}^n$ closed, then any continuous $f:B\to \partial [0,1]^2$ admits an extension

Prove or refute: Let $A$ be a closed subset of $\mathbb{R}^n$, for some $n$, and $B$ be a closed subset of $A$. Then any continuous function $f:B\to \partial[0,1]^2$, where $\partial[0,1]^2$ is ...
2
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1answer
77 views

Classifying continuous maps from closed 2-manifolds to various closed manifolds

I believe my question should be simple. The question is more physically oriented and originated from one of Witten's papers, "On Holomorphic Factorization of WZW and Coset Models", where he considered ...
2
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1answer
76 views

Show that if $\phi$ is a cocycle then $\phi(f\cdot g)=\phi(f)+\phi(g)$ for

This is an exercise from Hatcher: Let $X$ be a topological space, $G$ an abelian group. Regarding a cochain $\phi\in C^1(X;G)$ as a function from the paths in $X$ to $G$, show that if $\phi$ is a ...
3
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1answer
71 views

cohomology of total space

Suppose $\mu:E\to B$ is a fiber bundle with fiber $F$. Furthermore, $F$ and $B$ have vanishing odd dimensional cohomology group. Is it true that $E$ has vanishing odd dimensional cohomology group? You ...
5
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1answer
72 views

Non-existence of $C^1$ injective mapping $\mathbb{R}^3 \to \mathbb{R}^2$.

A friend of mine did a test yesterday where it asked to prove that there does not exist a $C^1$ injective mapping $\mathbb{R}^3 \to \mathbb{R}^2$. This is an immediate result from invariance of ...
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0answers
33 views

Framed nullbordant and calculation of framing in coordinate chart

To prove the Hopf degree theorem (theorem 2.37) D. Freed in his notes proves the following lemma I have two questions about this lemma: (1) how to calculate the framing at time $s$? what is the ...
3
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1answer
55 views

Stronger version of Acyclic Models Theorem

Let $\mathscr{C}$ be an abelian category. If $P_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C})$ is a bounded below complex of projectives, and $C_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C}...
2
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1answer
28 views

Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$

On page $10$ of Hatcher's Vector Bundles and K Theory, he gives a proof that the Whitney sum of the trivial line bundle over $\mathbb{R}P^n$ and the tangent bundle is equal to the Whitney sum of ...
1
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1answer
37 views

Construct a space with given fundamental group

I am trying to find out how to construct a space with the following fundamental group: $ \pi_{1}(X)= \langle a,b, c \mid b^{2}ac, c^{-1}a^{2} \rangle$ What is the main strategy for solving this kind ...
1
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0answers
36 views

Can all manifolds be framed?

In D. Freed's notes pp.6 he defines a framing of a submanifold to be a global trivialization of the associated normal bundle (tangent space of the manifold quotient out that of the submanifold, rather ...
2
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0answers
44 views

nef Line bundles over Kähler manifolds

I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let $X$ be a complete complex projective variety, and $L$ a ...
2
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1answer
50 views

Does a group action lifted to the universal cover commute with the fundamental group action?

Question: Let $\varphi \colon G \to \text{Homeo}(X)$ be a group action on a topological space $X$ with basepoint $x_0$ and universal covering $\pi \colon \widetilde{X} \to X$. Then the subgroup of ...
4
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2answers
91 views

Nonexistence of a continuous injection $f:S^2 \rightarrow \mathbb{R^2}$

What is the "easiest" way to show that there is no continuous injection $f:S^2 \rightarrow \mathbb{R^2}$? Sure the Borsuk-Ulam theorem implies that result, but this may be a "difficult" way.
0
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0answers
23 views

Is there a way to find the euler class of the tangent bundle of the sphere from the cohomology ring of real projective space?

So (I am pretty sure) that the tangent bundle over the sphere is the pull back of the tangent bundle of real projective space. I know that the sphere gives a double cover of real projective space and ...
2
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3answers
74 views

Constructing a map of degree 2 $f:T^2\rightarrow S^2$

I know the definition of degree and homology type stuff. But I don't know what a map $T^2\rightarrow S^2$ should actually look like. We never work with explicit examples in my class and I just have no ...
1
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1answer
42 views

Why is $F_{n,0}=H_n(K)$ for an arbitrary filtered complex?

Let $... \subset K_{-1}=0 \subset K_0\subset ...K_n \subset...$ be an arbitrary filtered chain complex with $colim_n K_n:=K$. Let $F_{p,p+q}=im(H_{p+q}(K_p) \to H_{p+q}(K))$ Mosher and Tangora ...
2
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0answers
30 views

Orientation of vector space given that of its subspace and the associated quotient

In D. Freed's lecture notes pp.2 he mentions the following way to define the orientation of a vector space given that of its subspace and the associated quotient. let $0 \to V' \to V \to V'' \to 0$ ...
2
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2answers
50 views

$\mathbb{R}^n - B[0,r]$ is simply connected if $n>2$

Question:$\mathbb{R}^n - B[0,r]$ is simply connected $\iff$ $n>2$. I have to prove or disprove. I know prove that for $n \in \{1,2\}$, $\mathbb{R}^n - B[0,r]$ is not simly connected. So I want ...
0
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1answer
26 views

Relative homology groups for torus with a point removed

I am trying to compute the relative homology $H_{n}(X, \partial X)$ for all $n \geq 0$, where $X$ is a torus with a point removed. $X$ is homotopy equivalent to wedge sum of two circles $S^{1} \vee ...
1
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0answers
34 views

Relation between nonorientability of the Möbius strip and the Möbius bundle

There are two ways in which the open Möbius strip $M$ is related to orientability: $M$ is nonorientable as a manifold; $M$ is the total space of the nonorientable line bundle $M \to S^1$. Is there ...
4
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0answers
96 views

Cup/cap product: sheaf cohomology vs singular cohomology

Is anyone aware of a good resource which deals with how the cup/cap products of sheaf cohomology classes are a generalization of those in singular cohomology? I would say that I already understand the ...
0
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1answer
33 views

Homology group versus group homology

If we have a simplical complex $K$, then we are able to define $C_i(K)$ as the free abelian group over $\mathbb Z_2$ with the basis of all $i$-dimensional simplices. By using the boundary map we are ...
0
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1answer
72 views

Prove that two $n$-sheeted covering space of $S^{1}$ are isomorphic.

We have a Blaschke product $B(z) \colon S^1 \to S^1$ of order $n$ and the map $f \colon S^1 \to S^1$, $f(z)=z^n$. Both maps are regular on $S^1$. We have already proved that both are $n$-sheeted ...
0
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1answer
35 views

Computing Fundamental Group of $S^1$

I have a hard time in understanding this from books and papers I found online. So anyone please suggest me some good reference or preferably an online lecture for computation of fundamental group of $...
1
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1answer
13 views

On linear homotopy of operators

Let $F$ be an isomorphism of euclidian space $E$, with orthonormal basis $\{e_1,\ldots e_n\}$. Let $F'$ be orthogonalised $F$. Is any operator $F_t$ from linear homotopy of $F$ and $F'$ an ...
3
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1answer
60 views

Jordan curve theorem for a polygon

Let P be a Jordan polygon on the complex plane. I would like to prove Jordan curve theorem for P using an elementary method. It seems that we can prove it using the winding number with respect to P. ...
0
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1answer
47 views

Triangle inside a simply connected open subset of the complex plane

Let $U$ be a connected open subset of the complex plane. Suppose $U$ is simply connected, i.e. its fundamental group is trivial. Let $T$ be a triangle whose boundary is contained in $U$. It is ...
1
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1answer
41 views

Computing the homology of the torus with coefficients in $\Bbb F_p$, using two methods

I have some trouble to compute the homology of the torus with coefficients in $\Bbb F_p$ for $p$ a prime number. In particular I have a problem for $H_1$ : 1) The first way to compute it is to use ...
6
votes
1answer
142 views

Does fundamental group distinguish between any two non homeomorphic topological space?

I am new to fundamental group. I was reading Munkres and found that need of fundamental group was to distinguish between non-homeomorphic topological spaces. So my question is, does fundamental ...
1
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0answers
25 views

Covering map, end points of two paths equal criterion

I was reading a proof of the following lemma: $\textbf{Lemma}$: Let $p\colon X \to B$ be a covering map and let $\gamma, \gamma'$ be two paths in $X$ beginning at $x_0$. Let $u=p\gamma$ and $u'=p\...
3
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2answers
56 views

Is there a retraction of $S^1\vee S^2$ onto $S^1$?

I am currently working through a problem that requires me to prove that $\pi_1(S^1\vee S^2)\simeq\mathbb{Z}$ directly by showing that the homomorphism induced by the inclusion of $S^1$ on $\pi_1$ is a ...
3
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1answer
43 views

Normal bundle of an embedding of a parallelizable manifold into a parallelizable manifold

This question is motivated by an observation of Milnor. Theorem: Let $M^m$, $N^n$ be parallelizable smooth manifolds, and $i:M\to N$ an embedding. If $n>2m$, the normal bundle is trivial. The ...
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0answers
13 views

Existence criterion of $Spin_{\mathbb{C}}$ structure

In deriving the existence criterion of $Spin_{\mathbb{C}}$ structure Ralph Cohen in his notes on the topology of fiber bundle (pp.169) uses the following diagram $\require{AMScd}$ \begin{CD} BSpin_{\...
0
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1answer
19 views

Explicit formulation of hermitian form and corresponding alternating form

I can't understand the following basic thing: Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\...
1
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1answer
23 views

Properly discontinuous group actions - Hausdorffness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
1
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1answer
63 views

Hatcher's exercise 1.2.22 on the Wirtinger presentation

Here exercise 1.2.22 is recalled, but the asker seems to know how to solve it assuming the "geometry is valid". I however, do not know to use the van Kampen theorem in order to find the relations $...
2
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0answers
51 views

Topology of CW-complex and attaching map

I think I must have a fundamental misconception in place right now in my mind. When defining a CW-complex, we use inductively continuous maps from $f_{\partial \sigma} :S^n \to K^{(n)}$. We then ...
4
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2answers
65 views

Why is $H_5(K(\Bbb Z_n,4))$ finite?

I want to see that the cohomology $H^i(K(\Bbb Z_n,4); \Bbb Z)$ starts with $\Bbb Z_n$ in degree 5. How do we know that $\operatorname{hom}(H_5(K(\Bbb Z_n,4);\Bbb Z), \Bbb Z)$ is zero? I.e. why is $H_5(...
1
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0answers
57 views

Local isometry between non-positively curved cube complexes

Let $X$, $Y$ be non-positively curved cube complexes and suppose there is a local isometry $X \to Y$. If $Y$ is special, then so is $X$. This is Exercise 4.32 in M. Sageev's notes "CAT(0) Cube ...