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

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invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
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1answer
13 views

On the relations between rank and torsion of homology and cohomology of a CW pair.

I am reading Massey's book on algebraic topology and on the chapter of universal coefficient theorem of cohomology, there is this exercise 4.1 that I don't know how to solve. Let (X,A)be a pair such ...
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2answers
30 views

Rigorous Covering Space Construction

Construct a simply connected covering space of the space $X \subset \mathbb{R}^3$ that is the union of a sphere and diameter. Okay, let's pretend for a moment that I've shown, using van Kampen's ...
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1answer
30 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
29 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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0answers
36 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
53 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?
2
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1answer
32 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
49 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
35 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 ...
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1answer
30 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
35 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 $\Sigma K(G,n) \to K(G,n+1)$. ...
3
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1answer
33 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
30 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
28 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 ...
3
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0answers
47 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) ...
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0answers
24 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
15 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
18 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 ...
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0answers
56 views
+50

Example 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
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 ...
3
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1answer
45 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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1answer
50 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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1answer
42 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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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 ...
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2answers
36 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
31 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 ...
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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
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 ...
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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 ...
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0answers
31 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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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 ...
0
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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
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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1answer
33 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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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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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
42 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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2answers
40 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 ...
0
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1answer
27 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
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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0answers
27 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
27 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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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
55 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
42 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 ...