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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65 views

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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84 views

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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52 views

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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32 views

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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57 views

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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24 views

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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43 views

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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118 views

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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104 views

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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62 views

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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85 views

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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53 views

Why Is the Induced Map Not Zero?

I am reading "Modern Classical Homotopy Theory" by Strom and have come across the following. We are given a fibration $F\rightarrow E\rightarrow B$. One then has two pushout squares: $$\require{...
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41 views

Finding a homotopy map

Let $K=\mathbb R^2\times (-\infty,0)\subset \mathbb R^3$, and let $Q$ be an open connected subset of $\mathbb R^2$. Is the fundamental group $\pi_1(Q\times [0,1)\cup K)$ trivial? And is it possible ...
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113 views

Question About Transgression

I have been working on this question here. Here is the setup: First, all cohomology groups are assume to be with $\mathbb{Q}$ coefficients. We assume that $H^*(K(\mathbb{Q},n))=\mathbb{Q}[x]$, with ...
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125 views

What's wrong with my argument to compute the vector bundles on $S^1$?

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don't know where did I make a mistake. So as circle is ...
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89 views

Question on Snake lemma

we have Short exact sequence of chain complexe $0\rightarrow C\xrightarrow[]{f}D\xrightarrow[]{g}E\rightarrow 0$ i want to prove that there existe a longue exact sequence of modules $$...\rightarrow ...
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249 views

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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103 views

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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35 views

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector $\hat{n}(\mathbf{x})=(n^x(\...
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35 views

Triangulation of $\mathbb{R}P^n$

How to give a tiangulation of $\mathbb{R}P^n$, for general $n>=2$? I know how to do this when $n=2$, but I don't know the general case. Also, I know the CW-structure of $\mathbb{R}P^n$, does this ...
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38 views

Group of continuous automorphisms of cylindrical plane is a lie group

How do I prove this theorem? Theorem: The group of continuous automorphisms of a cylindrical plane is a lie group. In this context, cylindrical means the Laguerre plane. I found a paper it ...
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63 views

Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

Charles Weibel writes in his survey of homological algebra Riemann de fined a surface $S$ to be $(n + 1)$-fold connected if there exists a family $C$ of $n$ closed curves $C_j$ on $S$ such that ...
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194 views

find the connected covering space of $\mathbb{R}P^2 \lor \mathbb{R}P^2$

This is a problem on Hatcher' book. How to find all the connected covering spaces of $\mathbb{R}P^2 \lor \mathbb{R}P^2$? I don't know where to start. Is there a general way to construct the ...
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261 views

Intersection of simply connected sets II

I read the following statement in the old question "Intersection of Simply-Connected Sets" (Intersection of Simply-Connected Sets): If $U$ and $V$ are simply connected and $U \cap V$ is path ...
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149 views

Cohomology ring of $U(n)$

As you know $$H^\ast (U(n);{\bf Z})=\bigwedge_{\bf Z}[x_1,x_3,...,x_{2n-1}]$$ where $|x_i|=i$ To prove this we use Leray-Hirsch Theorem for $$\tag{*}\ U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$$ ...
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29 views

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 | ||z_1||^2 + ||z_2||^2 = 1\}$, $D^2 = \{z \in \mathbb{C} | ||z|| \leq 1\}$.

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 | ||z_1||^2 + ||z_2||^2 = 1\}$, $D^2 = \{z \in \mathbb{C} | ||z|| \leq 1\}$. Let $X$ be the quotient of the disjoint union of $D^2$ and $S^3$ by the smallest ...
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39 views

Relative homology of interlevel set

Let us consider a function $f\colon \mathbb{R}^3\to\mathbb{R}$, $f(x,y,z) = x^3+y^3+z^3 - 5yz$. Can anybody drop a hint how to compute relative homology of interlevel sets with coefficients in $\...
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434 views

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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54 views

Definition of HEP correct?

In Algebraic Topology (Allen Hatcher, pg 14) I read a definition of HEP: ...Suppose one is given a map $f_{0}:X\rightarrow Y$, and on a subspace $A\subset X$ one is also given a homotopy $f_{t}:A\...
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30 views

Critical group and morse Lemma

I have this theorem with a part of it's prove I have two questions: 1) what is the spectral decomposition of A ? 2)How to see that $B_{\varepsilon}\cap f_0 = \lbrace x\in H,||x||\leq \varepsilon ,...
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Fiber product and $G$-invariant maps

Let $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ be the fiber product of unitary sphere $S^n$ and $\mathbb{R}$ over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $S^n$ by antipodal relation and on $\mathbb{R}...
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46 views

Geometric interpretation of $R$-orientations

For an $n$-dimensional manifold $M$ and a ring $R$ one can define $R$-orientability, e.g. by choosing local orientations, i.e. generators of $H_n(M, M - \{x\}; R)$ for every point $x$ in the manifold, ...
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48 views

Intersection between cycles and dominant maps

i must say in advance that i'm not very familiar with algebraic cycles and intersection theory, so i hope my question is not too trivial. Let $X$ and $Y$ be two K3 surfaces. Let $Z=\sum_{i=1}^{k}...
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55 views

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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39 views

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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269 views

degree of a continuous map

Let $f(z)$ be a polynomial with complex coefficients. $f$ can be extended to a continuous map $\tilde f: S^2 \rightarrow S^2$ ($\mathbb C \cup \{ \infty \} \cong S^2$). Prove that the degree of $\...
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47 views

Stiefel-Whitney classes as induced application in cohomology

Let $\xi=(E,p,X)$ a bundle over a paracompact space $X$. We can consider the exact sequence $$ 0 \rightarrow SO(n) \stackrel{i}{\rightarrow} O(n) \stackrel{det}{\rightarrow} \mathbb{Z}_2 \rightarrow ...
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59 views

Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.

Let $T$ denote the torus $S^1\times S^1$ and let $M$ denote the Möbius band. Let $C$ be a simple closed curve in $T$ which bounds a 2-disk. Form a space $X$ by identifying the boundary of $M$ with $C$ ...
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53 views

problem related to the Mayer-Vietoris Sequence

Let $D_{k}$ be the surface obtained by removing k small disjoint open 2-discs from the unit disc $E^{2}$. Show that $D_{k}\simeq G_{k}$, the k-leaved rose. Let $M_{k}$ be the surface obtained by ...
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32 views

continuous discrete open map and topological dimension

Is there anyone who can help me to answer this question : Let $\Omega$ be an open bounded and connected set of $\mathbb{R}^n$. Let $A\subset \Omega$ be a closed set of Lebesgue measure zero and whose ...
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111 views

Cohomology and 1-forms with compact support

I'm, having troubles with the following Let $U$ be a bounded open set in $\mathbb{R}^{2}$ such that $\mathbb{R}^{2}\setminus U$ has $n+1$ connected components. Prove that $\dim(H_c^{1}(U))=n$. I ...
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92 views

Canonical bundle and Möbius bundle

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} \...
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91 views

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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51 views

Prove homotopy of maps

I am studying for a qualifying exam and I am having difficulty on the following problem and I don't really know how to begin. Any help would be great. Let $M, N$ be compact manifolds and for all $t \...
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174 views

Weak deformation retraction

This is part of a problem from Hatcher's book. Let $Y$ be the subspace of $\mathbb{R}^{2}$ shown in the picture. Let $Z$ be the zigzag space of $Y$ indicated by the heavier line. Show that there is a ...
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102 views

Topolologics $(2n+1)$-dimensional Manifold with ball removed orientable?

Suppose you have a compact, orientable $(2n+1)$-manifold $M$. You take a neighbourhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So the the boundary of $M\...
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216 views

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 $S^1$....
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Covering space problem from an old Qual

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
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93 views

Reduced homology exact sequences

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, ...