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

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3
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1answer
45 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 ...
0
votes
1answer
55 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 ...
1
vote
0answers
64 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
votes
0answers
66 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) ...
0
votes
0answers
36 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 ...
1
vote
0answers
32 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 ...
0
votes
1answer
65 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 ...
7
votes
2answers
238 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
votes
1answer
116 views

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 ...
1
vote
1answer
61 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 ...
2
votes
0answers
42 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
votes
2answers
58 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 ...
1
vote
2answers
36 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
votes
2answers
100 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 ...
0
votes
0answers
34 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
vote
1answer
44 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 ...
0
votes
1answer
23 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 ...
0
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0answers
37 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
votes
1answer
29 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 ...
1
vote
1answer
46 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 ...
0
votes
0answers
39 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 ...
1
vote
0answers
29 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
votes
1answer
41 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
votes
1answer
46 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 ...
1
vote
1answer
54 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
votes
0answers
50 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 ...
1
vote
1answer
47 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 ...
3
votes
2answers
143 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 ...
0
votes
2answers
57 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
votes
1answer
30 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
votes
1answer
124 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: ...
1
vote
0answers
33 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 ...
0
votes
1answer
72 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
votes
1answer
126 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 ...
4
votes
1answer
53 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
votes
2answers
229 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 ...
1
vote
1answer
73 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 ...
-1
votes
2answers
49 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 ...
1
vote
1answer
67 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 ...
1
vote
1answer
38 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 ...
4
votes
2answers
155 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 $\mathbb{R}^3$: ...
0
votes
1answer
30 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 ...
0
votes
1answer
40 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 ...
1
vote
3answers
60 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
votes
1answer
42 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 ...
0
votes
0answers
52 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 ...
0
votes
2answers
36 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 ...
2
votes
0answers
78 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 ...
0
votes
1answer
30 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 ...
2
votes
1answer
39 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: ...