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

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1answer
20 views

Is every finite CW complex is homotopic to simplicial complex?

Is every finite CW complex is homotopy equivalent to a simplicial complex?
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0answers
12 views

Classification of compact 3-delta-complexes made of a single simplex

With a single 3-simplex (by identifying its faces in couples) it is possible to make 39 compact delta-complexes that can be grouped in 8 classes of complexes having the same homology groups (see ...
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1answer
40 views
+50

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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0answers
23 views

How is this simplex a subspace?

Let $e_1$, $e_2$, and $e_3$ be the standard basis vectors of $\mathbb{R}^3$. Then the standard 2-simplex, $ \triangle^2$, is of the form $$t_1 e_2 + t_2 e_2 +t_3 e_3$$ where $t_1 + t_2 + t_3=1$. ...
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1answer
47 views
+50

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
27 views

Is nerve theorem always true?

Is the nerve theorem true for not paracompact spaces? Background: Nerve theorem states that if $U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many ...
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0answers
46 views
+50

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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1answer
29 views

What is the map $\Sigma K(G,n) \to K(G,n+1)$?

Since $\Omega K(G, n+1)$ is a $K(G,n)$, we have a CW approximation/homotopy equivalence $K(G,n) \xrightarrow{\sim} \Omega K(G,n+1)$. The adjoint of this map is a map $\Sigma K(G,n) \to K(G,n+1)$. ...
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1answer
29 views

the fixed points of symplectic diffeomorphism

Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0=id$,$f_1=f\ne id$ be a Hamiltonian path on M generated by a Hamiltonian function F. Then how to prove that f has a ...
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0answers
41 views

Fundamental group of some disk quotient [on hold]

What is the fundamental group of $(D^2{/x\sim ix},0)$ with $D^2$ being the disk, $i$ the imaginary unit and $x\in S^1$. How can I compute this?
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0answers
24 views

Given a group $G$, the existence of a space such that $\pi_1(X)\simeq G$.

I'm having trouble understanding Corollary 1.28 of Hatcher which proves that for every group $G$ there is a space $X_G$ such that $\pi_1(X_G)=G$. Since $G$ is the quotient of a free group $F$, we ...
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1answer
27 views

Hatcher's formula in homotopy equivalence proof

In the proof that two homotopic maps induce the same homomorphism in homology, appears the formula (bottom of p. 112, Hatcher, Algebraic Topology): \begin{gather} P(\partial \sigma) = \sum_{i<j} ...
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1answer
41 views

Every Group is a Fundamental Group

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
3
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1answer
30 views

Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
2
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1answer
231 views

K-fold covering

I'd like some help with this homework: Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for ...
3
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0answers
44 views

Definition of the triad homotopy groups

Let $ \ n \in \mathbb{N}$, $n \geqslant 2$. Equip $\mathbb{R}^n$ with the usual euclidean norm $ \ \Vert . \Vert : \mathbb{R}^n \to \mathbb{R}$, metric and topology. Consider $ \ p_0 = (1,0,0,...,0,0) ...
3
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1answer
31 views

Map from $\mathbb{R}\mathbb{P}^{2}$ to Klein bottle homotopic to constant map

I'm currently struggling with the following exercise: Show that any continuous map $$f: \mathbb{R}\mathbb{P}^{2} \to K$$ where $K$ is the Klein bottle, is homotopic to a constant map. I know that ...
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1answer
28 views

Nulhomotopic map from $S^1 \rightarrow \mathbb{C} - \{0\}$

Hullo, I am aware that the inclusion map $i : S^1 \rightarrow \mathbb{C} - \{0\}$ is not nulhomotopic since there is a retraction from $\mathbb{C} - \{0\}$ to $S^1$ making the induced homomorphism ...
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0answers
22 views

n-connected pair (X,A) implies that the inclusion A to X is an n equivalence

I apologize if this is a novice question but I wanted to get some details on this concept. The algebraic topology book I am reading says that a relative CW complex $(X,A)$ is n-connected when ...
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0answers
14 views

What is the “product rule” for the boundary map of a product of CW-complexes?

I'm reading some notes that compute the action of the boundary maps on the CW-complex $\mathbb{R}P^2\times\mathbb{R}P^2$. I know the chain maps for $RP^2$ itself are $d_0,d_1\equiv 0$, and $d_2=\cdot ...
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0answers
15 views

Injectivity of a map from a homotopy set to a homology group

Let $\Sigma$ be a closed Riemann surface and $X$ a 1-connected topological space. I would like to prove the following fact. (A) The map $[\Sigma,X] \to H_2(X;\mathbb Z)$ defined by $[f] \mapsto ...
7
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5answers
221 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
2
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0answers
25 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 ...
0
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1answer
218 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 ...
3
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1answer
44 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 ...
0
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1answer
46 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 ...
1
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1answer
41 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
35 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
28 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 ...
3
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2answers
46 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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0answers
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 ...
1
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1answer
19 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
14 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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0answers
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 ...
0
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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
24 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
29 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
24 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
25 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
33 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 ...
3
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0answers
38 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 ...
7
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1answer
79 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
292 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 ...