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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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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_{\...
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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 $\sum_{i=1}^ndz_id\... 1answer 23 views Properly discontinuous group actions - Hausdorffness I was told to prove the following: If an action is free and satisfies that each point has a neighborhood$U$satisfying$U \cap gU=\emptyset$except for finitely many$g\in G$, and moreover the space ... 1answer 63 views Hatcher's exercise 1.2.22 on the Wirtinger presentation Here exercise 1.2.22 is recalled, but the asker seems to know how to solve it assuming the "geometry is valid". I however, do not know to use the van Kampen theorem in order to find the relations$...
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 ...