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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4
votes
2answers
169 views

Broken line is NOT diffeomorphic to the real line

This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ...
2
votes
1answer
34 views

Surjective map between fundamental group of surfaces

Let $f\colon S_m\rightarrow S_n$ be a continuous map of degree $\pm1$. Then the induced morphism $f_\bullet \colon \pi_1(S_m) \rightarrow \pi_1(S_n)$ is onto. How can I prove this? I know that the ...
1
vote
1answer
43 views

Second homotopy of $S^1\vee S^2 \vee T^2$

How can I prove that the second homotopy group of $S^1\vee S^2 \vee T^2$ is infinitely generated? I know that the second homotopu groups of $S^2$ and $T^2$ are finitely generated, so is kind of ...
4
votes
0answers
63 views

example of toric varieties with nontrivial first cohomology group

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ...
1
vote
1answer
39 views

using algebraic topological method to solve a complex analysis problem

Let $f \colon D^{2} \to D^{2}$ be a continuous function, and view $D^{2}$ as the set of complex numbers with norm $\leq 1$. Assume that on the boundary, $f \colon \partial D \to \partial D$ is given ...
1
vote
0answers
34 views

Calculate fundamental group and construct covering space

Calculate the fundamental group of the tensor product sign,$\otimes$,a subset of a plane,based at the point in the middle of the sign.Exhibit a connected 3- fold covering of this space. I want to ...
-1
votes
1answer
62 views

Question about the notation $X/A$ in topology

In Hatcher, the notation $X/A$ as appearing in the following text is never defined: If $(X,A)$ is a CW Pair consisting of a cell complex $X$ and a subcomplex $A$, then the quotient space $X/A$ ...
1
vote
4answers
45 views

Real Projective $n$ space $\mathbb{R}P^{n}$

In example 0.4 of Hatcher, he says that $\mathbb{R}P^{n}$ is just the quotient space of the sphere $S^{n}$ with antipodal points identified. He then says that this is equivalent to the quotient of a ...
3
votes
2answers
47 views

A question on the non trivial rank three bundle on $S^2$

We know that number of rank $3$ vector bundles on $S^2$ is just the number of equivalence classes of maps from $S^1$ to $SO(3)$. This implies that there is one and only one non trivial vector bundle ...
1
vote
1answer
55 views

On the definition of projective vector bundle.

Definition 1 : Let $B$ be a topological space. A complex vector bundle of rank $n$ over $B$ consists of the data $(E,B,\pi,\{U_i\}_i,\{\phi_i\}_i)$ where $E$ is a topological space, $\pi:E\to B$ is a ...
1
vote
0answers
27 views

Geometric construction of $J$-homomorphism

In D. Freed's notes eqn (5.32), he defines the $J$-homomorphism geometrically by considering the equatorial $n$-sphere as an $n$-submanifold of $S^m$, and giving it a framing that makes it null-...
6
votes
1answer
88 views

what is the nature of a ball that goes over a “corner” of the real projective plane?

I'm make a little computer program to help me understand different 2d topological spaces, (such as torus and mobius band). I'm having issues with drawing balls that go over a corner of the real ...
0
votes
1answer
32 views

Tensoring over the group ring versus tensoring over the ring in view of group representations.

I was reading a chapters homology with local coefficients. Where one of the preliminary sections asks us to compute $$\mathbb{Z}_{+} \otimes_{\mathbb{Z}[\mathbb{Z}/2]}\mathbb{Z}_{-}$$ Here $\mathbb{...
0
votes
1answer
34 views

bottom map of pullback square is cofibration $\Rightarrow$top map is cofibration

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I am trying to show that for a given fibration $E \xrightarrow{p} B$, and a cell structure $\{B_n\}$ on $B$ that , that $p^{-1}B_n \to p^{-1}B_{n+1}$ is ...
3
votes
1answer
66 views

Vector bundles and de Rham cohomology

So $M$ is a compact manifold and I am asked to either prove the following statement or give a counterexample: if $\pi: E \rightarrow M$ is a vector bundle, then $H^2(E) \simeq H^2(M)$. I know the ...
1
vote
1answer
37 views

How do these sections arise in a Bott tower?

A Bott tower of height $n$ is a sequence of $\mathbb CP^1$ bundles $\require{AMScd}$ \begin{CD} B_n @>{\pi_n}>> B_{n-1} @>{\pi_{n-1}}>> \cdots @>{\pi_2}>>B_1@>{\pi_1}...
0
votes
1answer
32 views

Partial Converse to “Pushout of a cofibration is a cofibration”

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I want a converse of this fact specialized to the case where I am pushing out a map BY a fibration: I.e., if I am given a diagram $ \begin{CD} E_1 @&...
1
vote
1answer
33 views

A problem on finding some covering space

Describe three pairwise non-homemorphic two-fold coverings of $RP^{2}\vee S^{1}$. $RP^{2}$ is the real projective plane and $\vee$ represents the wedge product of topological spaces. I know that map ...
1
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0answers
61 views

how this two space are homotopy equivalent?

May be this is a very silly question but it is somehow not clear to me.... If we take the space sphere with a diameter attached between north pole and south pole then if we start sliding one point ...
3
votes
2answers
35 views

Suppose $M$ has trivial 1-st de Rham cohomology group. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$?

Let $M$ be a compact oriented smooth $n$-manifold, with $H_{dR}^1(M)=0$. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$? I know that if $M$ is simply-connected, we ...
0
votes
1answer
49 views

Connectedness of suspension of a topological space

The suspension $\Sigma X$ of a topological space $X$ is defined as the quotient space $$ \Sigma X=\dfrac {X\times \left[0,1\right]}{\sim}$$ where $(x,t)\sim (y,s)$ if and only if $s=t=0$, or $s=t=1$, ...
4
votes
2answers
51 views

Quotient of $\mathbb{R}^n$ with an unbounded equivalence class homeomorphic to $\mathbb{R}^n$?

Let $X=\mathbb{R}^n$. Suppose $X$ has an equivalence relation $\sim$ such that at least one class consists of a line (a $1$-D subspace) through the origin. If $X^*=X/{\sim}$, is it possible for $X$ to ...
1
vote
1answer
38 views

First Betti number definition

I found in the electric engineering literature this alternative definition of the first Betti number of an open set $\Omega\subset\mathbb{R}^3$ with Lipschitz boundary. $n_\Omega$ is the first Betti ...
0
votes
0answers
22 views

how to show a homotopy equivalent

I have to show that $Z=\{(x,y,z)\in\mathbb{R}^3\mid y^2>4xz\}$ has the same kind of homtopy as $S^1$. I'm try somethings, but they lead to nothing. I was able to prove that $Z$ is path connected. ...
1
vote
1answer
27 views

Connected CW-complex which is not locally finite

I am working on Topological Complexity of robot motion planning. I am looking for a connected CW-complex which is not metrizable. I have found that: Proposition 3.8. A connected CW complex $X$ is ...
2
votes
1answer
39 views

On Steenrod's realization of cycles problem.

There is old problem of realization homology classes of (closed) manifold $M^n$ by fundamental classes of its submanifolds. Partially it was solved by René Thom in his "Quelques propriétés globales ...
0
votes
0answers
20 views

Descending group actions to coverings

Let $X$ be a path-connected space with universal cover $\widetilde{X}$, let $Y$ be another covering of $X$ $$ \widetilde{X} \hspace{1cm} \\ \searrow \\ \downarrow\hspace{.5cm} Y\\ \hspace{.25cm}\...
1
vote
1answer
34 views

Finding cohomology group of open dense subset of Schubert variety

Let $Y=Gr_{m}(\mathbb{C}^n)$ be the Grassmannian of $m$-plane inside $\mathbb{C}^n$. Let $X$ and $X'$ be two Schubert varieties inside $Y$ such that $X'\subset X$ and $dim(X')<dim(X)$. Let $Z=X\...
1
vote
0answers
63 views

Determining if Algebraic varieties are homeomorphic

So attempting to use the language of algebraic geometry, and algebraic variety $V$ is a the set of points that is the solution to some collection of algebraic relations $$x_1, \ldots, x_n \ \text{s.t....
0
votes
0answers
28 views

Hopf fibration homeomorphism injectivity

I was wondering if there is a possibility to proof the homeomorphism between the 1-dimensional complex projective space $\mathbb{C}P^1$ and the 2-sphere $S^2$ via the Hopf fibration (I know already ...
1
vote
2answers
67 views

What is the kernel of the map $H_i(X;\mathbb Z)\to H_i(X;\mathbb Z_2)$?

Let $X$ be a topological space. By Universal Coefficient Theorem for Homology we have the exact sequence $$0\to H_i(X;\mathbb Z)\otimes\mathbb Z_2\to H_i(X;\mathbb Z_2)\to \text{Tor}_1(H_{i-1}(X;\...
1
vote
0answers
27 views

Coverings of topological spaces, wedge sum of compact real surfaces

How to determine all the coverings of the topological space obtained connecting a torus $\mathbb{T}$ with the real projective plane $\mathbb{P}^2$ in such a way that their intersection is only one ...
0
votes
0answers
29 views

The claim that $A \to X$ a cofibration implies $A \times I \to M_{A \to X}$ is an inclusion.

Akhil Matthew claims in https://amathew.wordpress.com/2010/10/07/cofibrations/ in parenthesis that given a cofibration, $A \xrightarrow{i} X$, the map $A \times I \to M_i$ into the mapping cylinder, ...
2
votes
2answers
51 views

There is no retraction of the solid torus $S^{1} \times D^{2}$ onto the torus $S^{1} \times S^{1}$.

I'm trying to use the fact that I know that there is no retraction of $D^{2}$ onto $S^{1}$ (since there can be no injection $\pi_{1}(S^{1}) \rightarrow \pi_{1}(D^{2})$) to show that if there were a ...
0
votes
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
votes
1answer
72 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
vote
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
vote
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
vote
0answers
45 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
votes
1answer
62 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
votes
1answer
64 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
votes
2answers
151 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
votes
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
votes
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
votes
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
votes
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
votes
1answer
74 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
votes
1answer
74 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
votes
1answer
68 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
votes
1answer
69 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 ...