Tagged Questions

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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $X/A$ ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $s=t=1$, ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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;\...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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$B\subseteq A \subseteq \mathbb{R}^n$ 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 ...
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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 ...
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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 ...
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 ...
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 ...