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

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A question on Kronecker Index

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to ...
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0answers
8 views

Exemple about the difference between Morse and degree theory

i found this example but i don't understand how we applyed Morse theory and why we can't applyed degree theory. if the functional $f$ behaves like $<lu,u>$ at infinity where the symmetric ...
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0answers
21 views

Path Homotopy in a Topological Annulus

Let $C_1$ and $C_2$ be simple, closed curves in $\mathbb{R}^2$ such that $C_1$ lies in the region bounded by $C_2$, and the origin $O$ lies in the region bounded by $C_1$. Define an annulus $A$ as the ...
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1answer
216 views

Computing the homology of S^2 via Mayer-Vietoris

I'm trying to compute the homology of the 2-sphere. I start by decomposing the sphere into a northern hemisphere and southern hemisphere, denoted by A and B, respectively, and allow these two to ...
2
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1answer
62 views

Question about Lifting of Maps in the Circle

Let $S^1 = \{z\in\mathbb{C}:|z|=1\}$. For all $n\in\mathbb{N}$, define $f_n: S^1\to S^1$ by $f_n (z) = z^n$. Given $n\in\mathbb{N}$, for what values of $m\in \mathbb{N}$ there exists a lifting of ...
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1answer
42 views

Understanding Hatcher's proof of Borsuk-Ulam theorem for $n=2$

I am trying to understand the proof of the Borsuk-Ulam theorem for $S^2$ given in Hatcher's "Algebraic Topology" (Th. 1.10), as another person does here, but we are stuck at different places, so I ...
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26 views

Cohomology class with trivial restriction to a very general fiber

Let $f:X\to S$ be a flat morphism of smooth complex projective varieties. Let $s\in S$ be a very general point. Suppose that $\omega\in H^{p,p}(X)$ is a cohomology class such that $\omega|_{X_s}=0\in ...
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1answer
41 views

Proof of FTA from Hatcher

This is the proof of the fundamental theorem of algebra (FTA) given in Hatcher's Algebraic Topology textbook (I have underlined the relevant part): Could someone explain why $r$ needs to be ...
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1answer
36 views

Homeomorphism - transforming mug into donut

I read that a map is 'visually' a homeomorphism if you don't have to fold or tear the object. Thus, I was wondering what the problem with folding is? I guess that in this statement they don't assume ...
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2answers
30 views

connected sum of two surfaces

I was reading Massey's textbook on Algebraic topology and the author claims that if $S_2$ is a 2-sphere then $S_1 \# S_2$ is homeomorphic to $S_1$. I don't know why that is true and since I'm very ...
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2answers
27 views

Description of real projective spaces in various contexts

What I want to know is : What is the description of real projective spaces (specially $RP^0$, $RP^1$, $RP^2$) respectively in context of topology, geometry and algebra? I'm searching for simple ...
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2answers
44 views

Homeomorphic, homotopy equivalent and deformation retracts. How do I get a feeling for this?

We have homeomorphism, homotopy equivalences and deformation retracts ( which are a particular case of the latter). Now my problem is that I know what they all mean, but I have troubles to see them ...
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15 views

showing that n-fold projective plane is homeomorphic to (n-1)/2T#P or (n-2)/2T#K

I solved it by using first homology groups..but the instructor told me to prove it just by cutting and pastnig and some inductive method.... Could anyone show me how to show n-fold projective plane ...
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1answer
17 views

Mapping torus of Klein bottle, from discussion in Hatcher p. 152.

At the very bottom of page 151 to the top of 152 in Algebraic Topology by Hatcher, it says In the case of the mapping torus of a reflection $g:S^1\to S^1$, with $Z$ a Klein bottle, the exact ...
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1answer
237 views

Klein bottle covered by the torus

Maybe this is an idiot question and I'm missing something very trivial. This question question was asked here before, but the answer (which apparently is equal to the one that I created) seems ...
0
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1answer
9 views

How do you compute the simplicial homology of an $n$-gon with all edges and vertices identified?

Suppose you have an $n$-gon with all vertices identified, and all edges identified. I think the optimal way to compute the homology groups would be to view this as a cell complex consisting of a ...
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14 views

What does it mean to say a simplex has all its vertices distinct?

In Hatcher I'm trying to solve the problem: Show that the second barycentric subdivision of a $\Delta-complex$ is a simplicial complex. Namely, show that the first barycentric subdivision produces a ...
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1answer
25 views

Could anyone suggest me a counter example about liftings?

A book reads: Refer to the proof of the following assertion: Given a map $F\colon Y \times I \to S^1$ and a map $\tilde{F}\colon Y \times\{0\} \to \mathbb{R}$ lifting $F| Y\times\{0\}$, there is a ...
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1answer
22 views

Local degree of local homeomorphism is $\pm 1$

Let $f:X\to Y$ be a local homeomorphism. I claim that local degree of $f$ is $\pm 1$. I was wondering if my proof is correct: Let $x\in f^{-1}(\{y\})$ , $U$ be a neighbourhood of $x$ and $V$ be a ...
4
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1answer
35 views

$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$ isomorphism in algebraic topology

In algebraic topology we have the result $$H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}.$$ In Massey's book, this is a result that follows from the fact that the sequence $$0 \rightarrow ...
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0answers
24 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
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1answer
22 views

Basic idea for finding critical point via Morse theory

Please what is the basic idea for finding critical point via Morse theory and critical groups? Thank you
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0answers
24 views

homotopy group of the limit space

Let $V_k(\mathbb{R}^{n+k})$ be Stiefel manifold. Using $\pi_i(V_k(\mathbb{R}^{n+k}))=0$ for all $i\leq n-1$, how to obtain $\pi_i( V_k(\mathbb{R}^{\infty}))=0$ for all $i\in \mathbb{N}$? Can I just ...
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1answer
32 views

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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1answer
32 views

Can the winding number be a non-integer?

The formal definition of a winding number: For a continuous loop $\gamma\colon[\alpha,\beta]\to\mathbb{C}\setminus\{a\}$ which doesn't pass through a point $a$, one has the function ...
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1answer
35 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 ...
3
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0answers
37 views

Homology of stunted infinite real projective space

Consider the following composite based map $$f: S^2 \xrightarrow{\sim} RP^2/RP^1 \to RP^\infty/RP^1$$ induced by the inclusion of the real projective plane $RP^2$ into infinite real projective space ...
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2answers
181 views

On the Homology of Posets

Is there a homology theory of posets which computes topological invariants (e.g., number of $k$-faces, etc.) of the associated Hasse diagrams (viewed as simplicial/cellular/singular complexes) as ...
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2answers
39 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
78 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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1answer
26 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 ...
9
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1answer
291 views

What is known about $\operatorname{Aut}(\mathbb{I}^n)$

A few months ago, I asked a related question: Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$? It was interesting for me to know that ...
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0answers
22 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
26 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 ...
3
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2answers
74 views

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, ...
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0answers
31 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', ...
3
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1answer
40 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 ...
3
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1answer
54 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 ...
5
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0answers
177 views

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
58 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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0answers
28 views

An Advice Concerning Master's Programme [closed]

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) ...
3
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2answers
46 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 ...
3
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1answer
226 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
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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 [closed]

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
52 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
32 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
29 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
42 views

Can we create a metric space of statements? [closed]

A metric space where each statements are like points and proofs are like lines.
5
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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 ...