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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A circle with a line bisecting it is homotopic to the wedge sum of two circles

Let $C$ be the circle centered at $0$ with radius $2$. $L$ is the segment connecting $(0,2)$ and $(0,-2)$. Prove that $F = C \cup L$ is homotopic to the wedge sum of two circles. Intuitively, I ...
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31 views

Homology Poincare Homology Sphere by Mayer-Vietoris

I am working through some pages of Dale Rolfsen, Knots and Links, AMS Chelsea Publishing, 2003, in order to understand Dehn approach to the original Poincaré conjecture. To be concrete with what I ...
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2answers
50 views

Fundamental group of a wedge sum, in general (e.g. when van Kampen does not apply)

What is the fundamental group of a wedge sum in general? e.g. including the times when van Kampen cannot help us. The Wikipedia article on wedge sums mentions that Van Kampen's theorem gives ...
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1answer
18 views

Show that the maps are chain homotopic

Let $\Delta _{2}$ be a 2-simplex, $I=\left [ 0,1 \right ]$. Given are two maps $i_{0}:\Delta _{2}\rightarrow \Delta _{2}\times I$, defined by $x \mapsto (x,0)$ and $i_{1}:\Delta _{2}\rightarrow ...
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1answer
41 views

Deduce that there are short exact sequences

Show that for $n>0$ there is a short exact sequence of chain complexes $0\rightarrow C_i(X;\mathbb{Z})\stackrel{f}{\rightarrow} C_i(X;\mathbb{Z})\stackrel{g}{\rightarrow} ...
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0answers
37 views

limit of covering spaces

Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, ...
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27 views

$\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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0answers
28 views

Euler characteristic is the alternating sum of dimensions of the homology

Give a direct proof that Euler characteristic is the alternating sum of dimensions of the homology, using the rank of the boundary maps. This is my instructor's question of today's topology ...
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1answer
35 views

Show that $\Bbb R^{2n+1}$ is not a division algebra over $\Bbb R$ for $n>0$

This is an exercise from Hatcher's Algebraic Topology (exercise 2.B.8). Here is the problem statement: Show that, for $n>0$, $\Bbb R^{2n+1}$ is not a division algebra over $\Bbb R$ by showing ...
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0answers
32 views

Proposition of CW complexes

I am trying to prove the following result: Suppose $X$ is a connected finite CW complex. Fix a vertex $x\in X^0$. Prove that the inclusion $X^2\subset X$ induces an isomorphism ...
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1answer
62 views

Show that Riemann Surface is connected?

I was reading Artin's Alegbra when this question came into my mind. Consider $f(t,x)=x^{2}-t$ , The locus X of zeros in $\mathbb C^{2}$ of a polynomial is called Riemann surface of f. I understood ...
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49 views

Quick question: Line bundle on union of two lines

Let $l_1$, $l_2$ be two lines in $\mathbb{P}^n$. What is the meaning of $\mathcal{O}_{l_1}(a_1)\cup\mathcal{O}_{l_2}(a_2)$ as a sheaf on the union $C=l_1+l_2$ of two distinct lines and why do we ...
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0answers
17 views

CW-complex via composition of pushouts and there characteristic maps

Strom defines CW-complexes in his book Modern Classical Homotopy Theory (p. 47, ch. 3.2.1) via composition of pushouts, i.e. given a discrete topological space $X_0$, he constructs $X_{n+1}$ from ...
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1answer
16 views

Loops homotopic relative to A

Let $(X,A)$ be a pair of path-connected spaces, where $A\subset X$. How do we see/prove that this statement is true? Two loops $\gamma_0,\gamma_1\in\pi_1(X,x_0)$ are homotopic relative to $A$ if ...
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2answers
64 views

Fundamental groups of path connected subspaces

Does every path connected subspace of $\mathbb{R}^2$ have a fundamental group the trivial or an infinity group? For example, for convex subspaces we know that, but if we take only path connected ...
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2answers
38 views

Contradiction in spectral sequence calculation of $H_*(BO(2))$

$\newcommand{\Z}{\mathbb{Z}}$ For this post I am going to assume the answer namely $H_*(BO(2))=\Z_2[w_1,w_2]$. Consider the fibration $S^1 \hookrightarrow BO(1) \to BO(2)$. The $E^2$ page has ...
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1answer
32 views

Questions on CW complex structure

REMARK: I had already posted these questions, about one hour ago, but one of the questions was not what I meant. I am in the beginning of my studies in Algebraic Topology and am studying CW complexes ...
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1answer
35 views

Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
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1answer
58 views

Local coefficients involved in the obstruction class for a lift of a map

I'm interested in understanding the importance of the local coefficients in the definition of the obstruction cocycle for a lift of a map $f\colon X \to B$ along a fibration $p \colon E \to B$. I'm ...
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0answers
51 views

Finding a problem book in algebraic topology

I simply need book with problems solved with greatest explanation possible. I know about Hatcher and have a great lecturer, so I do not need theory. I need problems solved in detail.
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0answers
15 views

Deck group of a connected n-fold cover must have at most n elements

Let $p:Y\to X$ be an $n$-fold covering map, with $Y$ connected. Show that $Deck(p)$ has at most $n$ elements. My thinking was to prove this by contradiction, i.e. suppose we have distinct ...
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2answers
33 views

Deck transformation of the $n$-sheeted covering $\Bbb S^1 \to \Bbb S^1$

From Hatcher: For the $n$-sheeted covering space $S^1\to S^1$, $z\mapsto z^n$, the deck tranformations are the rotations of $S^1$ through angles that are multiples of $2\pi/n$. Why is this so? I ...
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1answer
30 views

retraction of two surfaces (Hatcher 3.3.13)

This is problem no 13 in page no 258 of Hatcher's algebraic topology: Let $M'_h$ be a compact subsurface of genus $h$ with a boundary circle ,so $M'_h$ is homeomorphic to $M_h$ with one open disc ...
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1answer
39 views

$\mathbb Q$ is totally disconnected. What is the open set in subspace of $\mathbb Q$?

I am trying to understand the proof that $\mathbb Q$ is totally disconnected. If $Y$ is a subspace of $\mathbb Q$ containing two points, $p$ and $q$, we can choose irrational a lying between $p$ and ...
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0answers
20 views

Hatcher's notation and definition for the boundary of a singular chain complex

I have a hard time following Hatcher in general, but especially his notation in this case with the bar (page 108 in my pdf copy): For the boundary of a singular $n$-simplex, $\sigma$,he writes: ...
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1answer
54 views

Every loop space $(\Omega Y,w_0)$ has the structure of an $H$-group.

The most important example of an $H$-group is the loop space $(\Omega Y,w_0)$ of any pointed space $(Y,y_0)$. Let $\mu:\Omega Y\times \Omega Y\to \Omega Y; \;\; \mu(\alpha,\beta)=\alpha \star\beta$, ...
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0answers
25 views

Readable references on gauge theory, knot theory and related

I have read the books Baez, Gauge theory knots and gravity and adams the knot book. When I try to follow the references there seems to be a big gap in readability and prerequisities required. Are ...
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1answer
33 views

Homotopy connectedness

I am trying to prove the following statement: Let $X,Y$ be two homotopy equivalent topological spaces. If $X$ is connected then $Y$ is connected. So far, this is my attempt: If $X,Y$ are homotopy ...
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0answers
30 views

Fundamental group of a covering space

I understand the correspondence between the subgroups of a fundamental group $\pi_1(X)$ and the covering spaces of $X$. However, I do not understand what is implied about the fundamental groups of ...
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1answer
41 views

Poincaré Duality in Middle Dimension

I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
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2answers
56 views

Non-compact 3-manifold with incompressible boundary

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times ...
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1answer
35 views

Sphere with four points deleted

What is the universal cover of the sphere with four points deleted and a non-trivial abelian fundamental group?
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0answers
31 views

a diffeomorphism on a 3-manifold

Assume we have an embedded torus $T=S^1\times S^1$ in $3$-manifold $M$. We construct a diffeomorphism $f$ of $M$ as follows: Take a neighborhood $T\times I$ of the torus and set ...
3
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0answers
33 views

Obstruction for extending a map along a CW-inclusion: sufficient conditions?

I'm interested in a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition ...
1
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1answer
28 views

Exercise with solution in Algebric topology

Can anyone suggest a collection of (solved) exercises in Algebric topology? Undergrad level, as I want to study on my own and take an exam, I found some lecturenote but I need to see some example or ...
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1answer
40 views

Is the quotient space obtained by identifying the poles of a sphere homeomorphic to a closed surface?

I'm interested in the quotient space of $S^2$ obtained by identifying the poles, and in particular whether it is homeomorphic to a closed surface. I'm pretty sure its homotopic to one, just by ...
1
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1answer
19 views

Prove continuity of a function from $S^n$ to $\Bbb R^n$

Given an interval $k$-coloring of $[0,1]$, define a function $f: S^k \to \Bbb R^k$ as follows ($S^k$ is the $k$-sphere). Let $x = (x_1,x_2,...,x_{k+1})$ be a point on the $k$-sphere $S^k$. Define $z = ...
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1answer
58 views

Algebraic Topology; Hatcher 2.23

Example 2.23 from Hatcher starts by stating... Let us find explicit cycles representing generators of the infinite cyclic groups $H_n(D^n,\partial D^n)$ and $\tilde{H}(S^n)$. Replacing $(D^n, ...
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1answer
66 views

Hatcher's proof of the van Kampen Theorem (injectivity of $\Phi$ – unique factorizations of $[f]$)

I am trying to understand the details of Allen Hatcher's proof of the Seifert–van Kampen theorem (page 44-6 of Algebraic Topology). My question is regarding the same part of the proof mentioned in ...
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0answers
30 views

$f: M \to Y$ $C^{1}$ map between manifolds $M$ e $N$ with dimension $m$ and $n$ respectively then $f$ is locally proper

We say that $f:f: M \to N$ is locally proper if for all $x \in M$ there exist $V \ni x $ open in $M$ such that $f|_{\overline{V}}:\overline{V} \to Y$ is proper. I know two ways to prove it. First, ...
0
votes
1answer
18 views

Mapping a circle to a point on a sphere

I'm trying to come up with an explicit homotopy for a circular loop about the equator of a sphere and the constant loop (1, 0, 0). The stereo graphic projection was used for this problem elsewhere on ...
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2answers
32 views

Are locally contractible spaces hereditarily paracompact?

The question title says it all. For the record, I have no reason to believe that this is true, but my question has a bit of a background. I am reading Ramanan's Global Calculus book because I am ...
2
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1answer
49 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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1answer
61 views

A remark on colimits, and the infinite grassmanian I didn't understand in class.

My friend was lecturing and he wrote down $\operatorname{colim}_n Gr_k^n$ to be the infinite Grassmanian $Gr_k^\infty$. Then my teacher said that the maps from $Gr_k^n \to Gr_k^{n+1}$ are not clear ...
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1answer
43 views

Find $\pi_1(S^3\setminus X)$ and the homology groups of $S^3\setminus X$.

Let $X=\{(z,w)\in \mathbb{C}^2|z^5+w^3=0\}$. a) Show that $\mathbb{C^2}\setminus X$ deformation retracts to $S^3\setminus X$. b) Find $\pi_1(S^3\setminus X)$ and the homology groups of ...
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1answer
27 views

If $T_1$ and $T_2$ are path connected covering spaces of $X$ then prove that in the following case $f:T_1\to T_2$ is surjective.

Let $p_1 :T_1 \to X$ and $p_2 :T_2 \to X$ be covering. $T_1, T_2$ are path connected spaces, $f : T_1 \to T_2$ is a continuous map, and $p_1=p_2 f$. Please prove $f$ is surjective. Note:This is an ...
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1answer
64 views

Real singular (co)homology of projective plane/Klein bottle without Mayer-Vietoris/Van-Kampen [closed]

I'm reading differential geometry books and trying to learn the singular (co)homology with real coefficients of the Klein bottle and projective plane by their fundamental rectangles. Here is the ...
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2answers
65 views

Covering spaces as fiber bundles

I am reading Steenrod. I think there is something wrong or sloppy in his definition of a covering space as a fiber bundle. He writes: A fibre bundle consists of: (i) A topological space $B$ (ii) a ...
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1answer
37 views

CW structures of identification spaces, fundamental groups and universal covers

I am working on this exercise concerning the relation between some identification spaces and their CW structure and I am a bit confused. First, the question is the following: For each identification ...
2
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1answer
20 views

Even dimensional Lens spaces

I am studying on my own Lens spaces from the algebraic topology viewpoint. I read about them on Hatcher's book as you can deduce from some of my previous questions: ...