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

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Follow-up to Previous Question on Klein Bottle

Here's the previous question: Homology of the Klein Bottle It asks what are the homology groups of the Klein bottle. My question is this: Are we always working over $\mathbb{Z}$? Say we denote by ...
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0answers
12 views

Codimension 1 homology represented by Embedded Submanifold

I'm looking for a reference for the following statement: Given an oriented manifold $M$ and a class $\xi \in H_{n-1}(M)$, there is some embedded oriented submanifold $F^+ \hookrightarrow M$ such that ...
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2answers
25 views

What is the universal cover of a discrete set?

Just curious, what is the universal covering space of a discrete set of points? (Finite or infinite, I'd be happy to hear either/or.) If there is just a single point, I think it is its own universal ...
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1answer
21 views

How does one compute a group modulo a torsion group?

Let's say I have some group $G$ and a subgroup $H$ such that $H$ is a torsion group (i.e. $\forall h \in H$, $h$ has finite order. How do I compute the factor group $\frac{G}{H}$? What effect does the ...
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1answer
56 views

Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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0answers
17 views

Orientability of space with only even dimensional cells.

I have the following question: Suppose a compact $\mathbb{R}$-manifold has finite cell decomposition with only even dimensional cells. Then $M$ is orientable. It's a theorem that a closed ...
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1answer
17 views

Finding simple homotopy type

I have an excercise that I kind of dislike: Given $T-\{p,q\}$ where $T = S^1 \times S^1 $ and $p,q \in T$ two different points, I am supposed to find a simple homotopy equivalent space by ...
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28 views

showing that maps from circle to circle are not homotopic

define fn by fn(e^ix)=e^inx for some integer n and every radian x then I want to show that if m, n are different then fm and fn are not homotopic. Here it does not just mean 'not path-homotopic', ...
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1answer
52 views

Constant maps induce zero homomorphism

It seems reasonable for me that if $f:X\rightarrow Y$ is the constant map then $f_{*}:H_{n}(X)\rightarrow H_{n}(Y)$ is the zero map for $n>0$. But I don't see how to prove this. If $n$ is odd then ...
3
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1answer
38 views

What does it mean that the quotient $S^n\to\mathbb{R}P^n$ acts as the identity on the upper hemisphere, and the antipodal map on the lower hemisphere?

I'm not sure how the degree of cellular maps are computed when finding the homology of $\mathbb{R}P^n$. I know $RP^n$ has CW structure with a cell in each degree, and $e^k$ is glued to $RP^{k-1}$ by ...
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0answers
27 views

An Advice Concerning Master's Programme [on hold]

Which of these programmes is a better choice, if one wants to pursue a degree in pure mathematics? (In Geometry, Topology and Algebra, in particular, algebraic geometry) 1) ...
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2answers
43 views

Fundamental group of two tori with a circle ($S^1✕${$x_0$}) identified

Compute the fundamental group of the space obtained from two tori $S^1✕S^1$ by identifying a circle $S^1✕${$x_0$} in one torus with the corresponding circle $S^1✕${$x_0$} in the other. Using van ...
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1answer
36 views

Construction of a covering space as a fibre bundle

In a direct proof of the equivalence of categories between the covering maps $p:(\hat X, \hat x) \rightarrow (X,x)$ of a topological space $(X,x)$ for sufficiently beautiful $X$ and the ...
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2answers
34 views

Wedge sum of spheres [on hold]

Let's $X$ be a CW-complex. If $X^{(n)}$ is the n-skeleton of $X$ and $\Lambda_n$ is a set of index. How could I prove that $X^{(n)}/X^{(n-1)}=\bigvee_{\alpha \in \Lambda_n} S^n_{\alpha}$? Thank you ...
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0answers
39 views

Question about Serre fibration.

I don't know what to do to prove the following result: $E=\{(x,y)\in \mathbb{R}^2:0 \leq y \leq x \leq 1\}, B=[0,1]:=I$ and $\pi:E \rightarrow B:(x,y)\mapsto x$, then $\pi$ has the homotopy lifting ...
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1answer
31 views

a question regarding ch.1 exercise in hatcher algebraic topology

the 4th problem in the p.38 of Hatcher algebraic topology says that when X is a union of finitely many closed convex sets, every path in X is homotopic in X to a piecewise linear path. But, a union ...
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1answer
26 views

Fundamental group of a kite shaped grid

What is the fundamental group of a "kite-shaped" two dimensional figure, with lines connecting opposite pairs of corner-points? (So, a diamond with a cross in the middle.) Doesn't this just ...
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1answer
41 views

Can we create a metric space of statements? [on hold]

A metric space where each statements are like points and proofs are like lines.
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2answers
42 views

Obtaining Wirtinger presentation using van Kampen theorem

Hatcher's Algebraic Topology, section 1.2, problem #22 describes an algorithm for computing the Wirtinger presentation of the complement of a smooth or piecewise linear knot K in $\mathbb{R}^3$: ...
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1answer
25 views

Graph as cell complex

I am reading a paper about graphs. In this paper, authors wrote "we view infinite graph as cell complex with usual topology". All graphs are infinite here. I know that we can consider graphs with ...
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1answer
32 views

homotopy type not constant during a homotopy

What is a possibly easy example of a topological space $X$ and a homotopy $H:X\times I\to X$, $H(x,0)=x$ for all $x\in X$, such that the homotopy type of the subspace $h_t(X)=H(X,t)$ is not constant ...
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3answers
50 views

how to show that a open set of $S^3$ is simply connected?

Let $B$ be the union of the compactification point and $(\mathbb R^3-X)$ in the one-point compactification of $\mathbb R^3$. (Here $X$ is a closed ball in $\mathbb R^3$.) Then I think $B$ is somehow ...
2
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1answer
26 views

Extension of a homeomorphism

Let X and Y Hausdorff normal topological spaces, and let N,M dense subspaces of X,Y, respectivaly. Let f a homeomorphism between N and M. Is true that exists an continuos extension F (between X and ...
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1answer
69 views

Main goal of algebraic topology and progress towards the main goal… [on hold]

In algebraic topology we use algebraic structure such as group to classify some topological spaces. To do that mathematician invented homotopy group,homology,co-homology etc.One can apply these tools ...
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23 views

How do i prove that this is homeomorphic to Klein Bottle?

My professor is teaching quotient space recently and he gives very informal arguments (drawing diagrams,arrows and stuff). I think he wants students to get familiar with quotient spaces and visualize ...
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2answers
30 views

A few questions regarding the winding number.

A few questions baout the winding numbers: Why do two homotopic paths have the same winding numbers? I think I can prove that two homotopic paths may have different winding numbers. Let $C$ be a ...
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0answers
37 views

Difference between two concepts of homotopy for simplicial maps?

I learn from Gelfand and Manin's Methods of Homological Algebra, Exercise 2 for I.4 that two maps $f,g\colon X\to Y$ between simplicial sets $X,Y$ are simply homotopic (maybe usually called simplicial ...
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1answer
20 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
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0answers
22 views

A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
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1answer
26 views

The signature of a product of surfaces

If $\Sigma_1$ and $\Sigma_2$ are surfaces (i.e. compact, oriented 2-manifolds without boundary), is the signature $\tau (\Sigma_1 \times \Sigma_2)$ well-known? Recall that the signature is the ...
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1answer
29 views

Fundamental crossed square of a square of spaces

I am a newbie at Topology and I didn't took the homotopy theory course. Sorry for that, but I need to know some facts about this paper ("Van Kampen Theorems for Diagrams of Spaces", authors R. Brown ...
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1answer
43 views

A condition for a covering map to be regular

We said that a path-connected covering map $p:E \rightarrow X$ is regular if: $\forall e \in p^{-1}(x_0): p_{\sharp} \pi_1(E,e)$ is a norm subgroup of $\pi_1(X,x_0)$. or equivalently: If closed ...
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1answer
28 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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1answer
41 views

Orbit space of $S^n \times S^n$ under the antipodal action

Write $S^n$ for the $n$-dimensional sphere, the space of vectors of length $1$ in $(n+1)$-dimensional Euclidean space. Consider the antipodal action on $S^n$, i.e. the action of $\mathbb{Z}_2$ given ...
4
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1answer
66 views

Fundamental group of the complement of an eight shape

What is $\pi_1(\mathbb{R}^3 \setminus (S^1 \vee S^1))$? I would say that the space deformation retracts to a sphere $S^2$ surrounding the missing eight with two sticks stuck in the two loops of the ...
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1answer
20 views

Why are degree maps for cellular boundary formula from $S^{n-1}\to S^{n-1}$?

For a CW-complex, there's the cellular boundary formula that $$ d_n(e^n_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_\beta $$ where the coefficients $d_{\alpha\beta}$ are the degrees of the map $$ ...
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1answer
15 views

If $D^1\cup_f D^1=S^1$?

Suppose $f\colon S^0\to S^0$, so we can form the attaching space $D^1\cup_f D^1$. Is my intuition correct that this space is just $S^1$? Since $S^0=\{1,-1\}$, $f$ is either the identity, or swaps the ...
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0answers
51 views

About homotopy fiber at Hatcher's book

What is the meaning of the statement: In this case a map $(I^{i+1},∂I^{i+1},J^i) \to (B,A,x_0)$ is the same as a map $(I^i,∂I^i) \to (F_f, \gamma_0)$ where $\gamma_0$ is the constant path at ...
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2answers
58 views

About the definition of homology

can someone explaine me this definition of Homology: "The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact." I know that homology measure the number ...
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1answer
45 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
3
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1answer
82 views

Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
3
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1answer
138 views
+50

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in ...
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2answers
34 views

How do we prove that the fundamental group is a group?

My understanding of the fundamental group is that it's the set of all loops starting and ending at a point $x_0$ in a space $X$, along with the operation of composition. For it to be a group, ...
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0answers
48 views

Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
3
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1answer
27 views

Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which ...
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2answers
21 views

retraction induced homomorphism is surjective

Im having a hard time proving this although it looks trivial... Let $r:X\to A$ be a retraction between a topological space $X$ and $A\subset X$ such that $r(a_0)=a_0$ for $a_0\in A$ then the induced ...
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0answers
40 views

Homomorphisms induced by maps $S^1 \times S^1 \rightarrow S^1 \times S^1$.

Problem 2.2.30 in Hatcher involves the homomorphisms $H_2(S^1\times S^1) \cong \mathbb{Z} \rightarrow H_2(S^1\times S^1) \cong \mathbb{Z}$ induced by The map $S^1 \times S^1 \rightarrow S^1 \times ...
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1answer
48 views

A question on finite non-contractible CW complexes

The algebraic topology book I am reading recently covered the following theorem named after Whitehead and corresponding direct consequence. THEOREM. If X is a CW complex of dimension less than n and ...
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1answer
67 views

Is the suspension of a countable collection of points in $\mathbb{R}$ a countable collection of circles?

I am extremely new to topology and taking an algebraic topology course, and I need some help understanding the behavior of suspensions. The problem I am working on asks about the suspension of the ...
2
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0answers
39 views

Homotopic paths implies equal winding numbers

I am trying to prove a proposition relating analysis and geometry. I have a general idea on how to prove it. However, a small part of the proof needs a lemma about path homotopy and winding number. ...