Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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13 views

What do you get if you glue a disk twice around a circle?

I would like to know what you get if you glue the disk $D^2$ around the circle $S^1$ via the map $\phi \colon \partial D^2\to S^1$, $\phi (e^{i\theta})=e^{2i\theta}$. I would have thought you would ...
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1answer
35 views

Cohomology Group of $CP^2 \wedge CP^2$

Calculate the cohomology group of $CP^2 \wedge CP^2$ To do this, at first I am trying to calculate the homology group and then use Universal Coefficient Theorem. To do this, at first I have ...
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1answer
39 views

What is the difference between CW-complex and Cellular complex?

Is every CW-complex is a Cellular space? Is its converse true? If it is true then what is the difference between them? We include the definition of CW-complex in algebraic topology given by ...
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0answers
33 views

Equivalent presentation for the fundamental group of the projective plane

We know that $\langle a,b;(ab)^2=1\rangle$ and $\langle z;z^2\rangle$ are presentations of the fundamental group of the projective plane. Therefore, one is obtained from the other via Tietze ...
3
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1answer
48 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
4
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1answer
42 views

Milnor's definition of bundle map in “Characteristic Classes”

In chapter 3 of "Characteristic Classes", Milnor defines bundle maps, requiring them to map fibers isomorphically onto fibers. Why not merely require homorphisms on fibers? (e.g., for the given ...
4
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0answers
33 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
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0answers
23 views

An example of $K(G,1)$ in Hatcher

A $K(G,1)$ space is a path-connected topological space $X$ with contractible universal cover and $$ \pi_1(X)=G. $$ I am reading about $K(G,1)$ spaces in Hatcher's textbook and I don't understand ...
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0answers
21 views

Immersion of punctured torus into Euclidean [duplicate]

(a) Show there is an immersion of the punctured torus $S^1\times S^1$ - {a point} into $R^2$. (b) generalized it to $T^n$ - {a point} into $R^n$ can you give concrete proof for these problem? ...
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1answer
49 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...
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1answer
27 views

Simple homotopy construction

I'm sure this isn't too difficult but i can't seem to do it if you have two loops $p_0 = e*g $ and $p_1 = g*e$ where $e$ is the trivial loop How would i construct an explicit homotopy between the ...
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0answers
22 views

Section of a covering projection from a connected space [duplicate]

Let $p:\overline{X}\rightarrow X$ is a continuous mapping. A continuous map $s:X\rightarrow \overline{X}$ such that $p\circ s =Id_X$ is called a section of $p$. Suppose $\overline{X}$ is connected ...
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1answer
18 views

The fundamental group of some wedge sum

I was wondering how one can compute the fundamental group of the wedge sum of a sphere and 2 circles , i know the fundamental group is Z*Z ,and that the fundamental group of a wedge sum is the free ...
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0answers
19 views

Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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0answers
19 views

Show that $p:SO_3 \to S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle

Show that $p:SO_3 \to S^2 $ defined as $p(A)=Ae_1$ is a fibre bundle. I know that $SO_3$ acts on $S^2$ transitively saying that $p$ is onto.I have a problem with local trivialization part.I have ...
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1answer
38 views

Homology of a simplicial set

Let $X$ be a simplicial set. Define the complex $(C^X_\bullet,D)$ by $$C^X_n=\bigoplus_{X_n} \mathbb{Z}$$ and $$D_n=\sum_{i=0}^n (-1)^i d_i:C_n \to C_{n-1}$$ where the $d_i$'s are the face maps. I ...
2
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1answer
27 views

Embeddability of connected sum of non-embeddable surfaces

Let $X$ be a surface which can not be embedded into $\Bbb R^n$. Let $X \# X $ denotes the connected sum of two copies of $X$. Then is it true that the connected sum $ X \# X $ is also not embeddable ...
2
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1answer
74 views

Examples with zero first Stiefel-Whitney class and nonzero second Stiefel-Whitney class

What's the simplest/most concrete vector bundle you can think of that has zero first Stiefel-Whitney class but non-zero second? That would be the simplest space that doesn't have spinors. (See Spin ...
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0answers
14 views

Morphism of modules of sections of pullback bundles

Suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles ...
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1answer
10 views

Section of pullback bundle

Suppose that $E \to B$ is a vector bundle and $f:A \to B$ is continuos. If $s$ is a section of $E$ how to define a section of pullback bundle? On wikipedia they say that it induces the section of the ...
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2answers
55 views

A problem on covering space from Hatcher book…

I was trying a problem from Hatcher's book Algebraic Topology, in section 1.3 problem number 12. Let $a$ and $b$ be the generators of $\pi_1(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. ...
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2answers
51 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
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0answers
29 views

“Cohomology classes correspond to homotopy classes of maps to Eilenberg Maclane spaces” and cup product?

I read this in Hatcher. I am especially interested in knowing if the cup product can be understood from this perspective? I would appreciate a reference.
2
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1answer
67 views

simply connected covering of a path connected space

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : ...
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1answer
27 views

About covering spaces

Suppose X is a topological space whose fundamental group is Z x Z x Z2 x Z3. Is it possible for the wedge sum of two circles to be a covering space for X? Can anyone help me with this ?
3
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0answers
37 views

Let $A$ be the annulus in $\mathbb{C}$, what is the space $A/{\sim}$ generated by identifying all points on the “inner circle” with each other?

The annulus is $A=\{z\in\mathbb{C}:1\leq |z| \leq 2\}$ and the "inner circle" here is the set of points $\{z\in\mathbb{C}:|z|=1\}$. If we identify all points of the inner circle together, then, ...
2
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0answers
45 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
2
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3answers
65 views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Taken from Hatcher, I think the free product $G =\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ should have infinite elements taking the form of words containing alternate elements, i.e. $a, b, ab, ...
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59 views

i want to know about cube root of unity [on hold]

Blockquotecomlex number cube root of unity origin of comlex numbers origin of cube root ofunity
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17 views

Questions about strata of a variety: the nilpotent cone of $\mathfrak{sl}_2$.

I am reading the lecture notes. In the end of page 3, let $X = \{(a, b; c, -a): a, b, c \in \mathbb{C}^3, a^2 + bc = 0\}$. It is said that there are two strata: the regular orbit $U$ and $0$. What is ...
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3answers
35 views

Relative homology $H_n(S^2,S^0)$, or other examples

I've been reading Hatcher and think I understand the idea of relative homology, but he only provides two (fairly trivial) examples, homology relative to a point computing $H(S^n)$ using $D^n$s. My ...
5
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1answer
64 views

fiber bundle in topological category and smooth category.

Let $M$ be a smooth manifold and $G$ be a Lie group. Denote by $Bun(M,G)$ the set of all equivalent smooth Principal bundle on $M$ with structural group $G$ in smooth category. And denote by ...
4
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3answers
65 views

The significance of filtered colimits in homotopy theory

I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask ...
3
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1answer
28 views

A question about the degree of a map

Consider $f,g: S^{n}\rightarrow S^{n}$ to be continuous maps. Seeing $S^{n}\subset\mathbb{R}^{n+1}$, let's say that $f$ and $g$ are orthogonal at $x\in S^{n}$ whenever $\langle f(x),g(x)\rangle = 0$ ...
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0answers
17 views

Conjugate group homomorphisms induce homotopic maps on classifying spaces

Let $G$ be a group and let $\phi: G \to G$ be the inner automorphism given by conjugation by an element $g' \in G$, i.e., $\phi(g) = g'^{-1} g g'$. I want to show that the induced map on classifying ...
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1answer
21 views

Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space?

Question. Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space? Let $p:E\rightarrow X$ be a covering space of X. I ...
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1answer
38 views

Fiber bundle beginner question.

I'm reading some notes on fiber bundles. Let $f:X \rightarrow Y$ be a continuous map of topological spaces. The author states: We say $f$ makes $Y$ a fiber space over $X$ if $f$ is locally trivial ...
4
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1answer
120 views

A question on Hawaiian earring

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group ...
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1answer
55 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
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1answer
88 views

How to show $S^n$ is not contractible without using Homology..

I know the prove $S^n$ is not contractible using homology.But I don't know how to prove it from definition of contractibility.Can anyone help me in this direction? Thanks.
7
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2answers
112 views

Fundamental group of a quotient on a solid torus.

It is easy to compute the fundamental group of a solid torus. You easily get $\mathbb{Z}$ just because the torus is the cartesian product of a circumference and a closed disk. The next step is ...
2
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1answer
72 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
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0answers
25 views

Finding whole number coordinates on continuous curves [on hold]

Imagine two surfaces in 3D space defined by known equations intersect and form a line in 3D. How could you find out if that curve formed by the intersection goes through any points where all 3 ...
2
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3answers
41 views

Difference between Wedge of countable infinite circle and Hawaiian ear ring?

Hawaiian ear ring is the union of countable circles at points (0,1/n) with radius 1/n.It seems to me that wedge sum of countable infinite circle is same as Hawaiian ring.But I found that this not ...
4
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1answer
61 views

Do join and suspension commute?

Do join and suspension of topological spaces always commute, i.e. is it true that $\sum(A\star B)=A\star(\sum B)$? I suppose that it is not true in general (but, for example, everything works in the ...
1
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1answer
65 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
2
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3answers
37 views

locally path connectedness

While studying covering spaces , hatcher mentioned the "shrinking wedge of circles" this space is locally path connected as I was told , but I wasn't able to prove it nor to see it, it looks like comb ...
2
votes
1answer
48 views

Calculating $H_1(\mathbb{R})$

Given the space $X=\mathbb{R}$, how can we calculate its first homology group $H_1(\mathbb{R})$? Intuitively, the object of first homology describes 1-dimensioal holes in the set which here doesn't ...
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1answer
44 views

classification theorem in a subset of R^2

I need some very simple results of algebraic topology but I am not sure where I can find them without having to swallow the whole theory. What I want: -An open bounded subset $A$ of $R^2$ is ...
3
votes
3answers
62 views

The biggest degree of a map between fixed surfaces

Let $S_1$ and $S_2$ be compact surfaces (real manifolds of dimension 2). None of them is a sphere. Question: What is the biggest degree of a smooth map from $S_1$ to $S_2$? Comment 1. I have a ...