# 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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### Maps $S^1 \to S^1$ of equal degree are homotopic.

Let $a(t)=e^{2\pi it}$ be the generator of $\pi_1(S^1,1)$ and we define degree of $f$ this way: $$[\omega^{-1}]\cdot f_*([a])\cdot [\omega]=\deg(f)[a]$$ where $\omega$ is any path from $f(a)$ to $1$....
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### On Dold fibration

The article of nLab on Dold fibration I have two questions: How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi:E \to B$ is a Hurewicz ...
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### Fundamental group of a complete intersection in real projective spaces

I'm trying to understand the fundamental group of the following complete intersection in $RP^2 \times RP^2 \times RP^1$: \begin{eqnarray} &&t_1 \left( x_1^3 + x_2^3 + x_3^3 + a x_1 x_2 x_3 \...
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### homeomorphism still isotopic to the identity after the deletion of points.

I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a ...
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### Simple question on splitting of cohomology groups.

From the exponential exact sequence, I have $$0 \rightarrow H^2(X,\mathbb{C})/H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{C}^\times) \rightarrow Tor(H^3(X,\mathbb{Z})) \rightarrow 0.$$ for some ...
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### Relative de Rahm cohomology computation for two disjoint circles embedded in $\mathbb{R}^2$

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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### HEP of subcomplex for product topology of CW-complexes

Suppose $X$ and $Y$ are CW-complexes and $A\subset X$ and $B\subset Y$ have properties such that the product topologies of $X\times B$ and $A\times Y$ are CW-complexes, such as when $A$ and $B$ are ...
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### What is the name of $C(A)/A$

Given a topological space $A$, $C(A)$ is the cone of $A$. The space $C(A)/A$ is clearly homotopic to the suspension. My question is if it has a widely known name?
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### Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
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### finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
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### Quotient of a Riemannian manifold by a non-free group action

Take the example of $\mathbb{R}^2$ acted on by $C_n$ via a rotation of an angle $2 \pi/n$ around the origin. The quotient is a cone whose apex $V$ is the image of the origin. I have two questions: ...
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### Working with $\bigotimes_{\mathbb{Z}_X}$?

What is the meaning of the tensor product sign in a formula such as $$A \bigotimes_{\mathbb{Z}_X} B$$ where X is a real manifold of arbitrary dimension? B is a orientation sheaf (ultimately a matrix ...
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### Genus of Fermat Curve

The genus of a projective Fermat curve $x^d+y^d=z^d$ in $\mathbb{P}^2$ can be computed using the formula $g={d-1\choose 2}$, where $d$ is the degree. Is the genus of the affine curve $x^d+y^d=1$ the ...
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### Connected sum of two surfaces is a surface?

IS the connected sum of two surfaces a surface? Im having hard time trying to see this. Can someone kindly help me? thanks.
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### Find all the covering spaces of the Klein Bottle

I am studying for a qualifying exam and it asks for all the covering spaces of the Klein Bottle. Does anyone have any suggestions? I am not sure how to go about finding ALL of them.
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### Vector field and a solution of an ODE

I have this: And my questions are : 1)what is :"The local theory of differential equations in a Banach space" 2)Why it's implies that each solution of (6) is equal to $\eta$ Please Thank you .
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### Question on critical groups

I have this theoreme with it's proof But i don't understand who is $f_0$ ? Can someone help me please ? Thank you .
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### Homotopy type of surface of revolution

Let $X$ be a finite graph lying in a half-plane $P\subset\mathbb{R}^{3}$ and intersecting the edge of $P$ in a subset of the vertices of $X$. Describe the homotopy type of the surface of revolution ...
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### Computing singular homology

could you help me solving this questions,please Let $D^2$ be a 2-dimensional disc and $M$ be the Möbius strip. Note that the boundary of both $D^2$ and of $M$ is homeomorphic to the circle $S^1$....
Suppose that S$^1 \times P^2$ covers some space, and let $h$ be a covering translation. Show that the induced isomorphism $h$ of $H_1(S^1, P^2)$ must be the identity
Given $H$ a homology theory, let $f: X \to P$ the unique map from $X$ to a one point space $P$. This induces a map $f_{*}: H_i(X) \to H_{i}(P)$. We define the reduced homology to be \$\tilde{H}_i(X, ...