# 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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### 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 ...
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### Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
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### 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. ...
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### Duality between Thom space and a manifold embedded into a sphere

In a document https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf (s. 19) it is mentioned that there is a map $S^n \to M^+ \wedge \mathrm{Th}\left(\nu \left(M, S^n\right)\right)$, which gives 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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### 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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### 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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### The homology of wedge sum

This is an exercise of Bredon (pg. 190) which I tried to do but got stuck at one part. He asks the following: Let $X$ be a Hausdorff space and let $x_0 \in X$ be a point having a closed ...
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### Why bother showing $S^{1}$ covers itself?

I've just been introduced to covering spaces, and one of the examples I've been shown is that $p: S^{1} \to S^{1}$, $p(z)=z^{n}$ is a covering map for every $n$. My question is: why would you care? ...
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### 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 ...
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### I want to prove that $B=S^2 \cap \{(x_1, x_2, x_3) : x_i \geq 0 \}$ is homeomorphic to the disk $B^2= \{ x \in \mathbb{R^2} : ||x|| \leq 1 \}$

I claim that the map $f:B \to B^2$ defined by $f(x_1, x_2, x_3)=(x_1, x_2, 0)$ is a continious, bijection with $g: B^2 \to B$, $g(x_1, x_2)=(x_1, x_2, \sqrt{1-(x_1^2+x_2^2)})$ it's inverse. So i ...
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### Does topological degree generalize to maps that aren't between closed connected orientable manifolds?

From what I gather, the degree of a map originally arose in the context of studying maps $f:S^n\rightarrow S^n$. Since $H_n(S^n)\cong \mathbb{Z}$, the induced map $f_\star$ has the form $x\mapsto kx$ ...
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### Pictures and illustrations of attaching a 2-cell to a wedge of circles?

I am having a lot of trouble visualizing how to attach a disk "along a word". Where can I find some pictures and illustrations of this procedure?
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### Lifting correspondence in Algebraic topology

Recently I have been studying algebraic topology and came across the notion of lifting correspondence. Here, lifting correspondence definition is the same that Munkres uses in his book. However, I ...
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### Prove of homotopy equivalence using differential equation

I have to prove that $R^3 \setminus L_1,...,L_n$, where $L_1,..,L_n$ are non intersecting lines, is homotopy equivalent to a wedge sum of $n$ circles. So once I've managed to show that it is possible ...
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### There is no antipode-preserving map $f:S^{n+1}\rightarrow S^n$

I am trying to prove that there is no antipode-preserving map $f:S^{n+1}\rightarrow S^n$. Here is what I have: Suppose there is an antipode-preserving map $f:S^{n+1}\rightarrow S^n$. If we restrict ...
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### First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
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### Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
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### reference for simplicial complexes and triangulation

I am trying to understand simplicial complexes and triangulations, am following Basic Algebraic topology by Prof. Anant R Shastri and not very comfortable with the way it has been explained. Suggest ...
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### Natural isomorphism $\tilde H_i(X) \xrightarrow{\cong} \tilde H_{i+1}(\Sigma X)$ where $\Sigma X$ is the suspension of $X$.
Define $\Sigma X$ to be the quotient space of $[-1,1]\times X$ obtained by identifying ${0}\times X$ and ${1}\times X$ to two points respectively. For any homology theory (satisfying Eilenberg-...
On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the ...