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

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On linear homotopy of operators

Let $F$ be an isomorphism of euclidian space $E$, with orthonormal basis $\{e_1,\ldots e_n\}$. Let $F'$ be orthogonalised $F$. Is any operator $F_t$ from linear homotopy of $F$ and $F'$ an ...
3
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1answer
72 views

Jordan curve theorem for a polygon

Let P be a Jordan polygon on the complex plane. I would like to prove Jordan curve theorem for P using an elementary method. I think that we can prove it using the winding number with respect to P. Am ...
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1answer
50 views

Triangle inside a simply connected open subset of the complex plane

Let $U$ be a connected open subset of the complex plane. Suppose $U$ is simply connected, i.e. its fundamental group is trivial. Let $T$ be a triangle whose boundary is contained in $U$. It is ...
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1answer
41 views

Computing the homology of the torus with coefficients in $\Bbb F_p$, using two methods

I have some trouble to compute the homology of the torus with coefficients in $\Bbb F_p$ for $p$ a prime number. In particular I have a problem for $H_1$ : 1) The first way to compute it is to use ...
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1answer
142 views

Does fundamental group distinguish between any two non homeomorphic topological space?

I am new to fundamental group. I was reading Munkres and found that need of fundamental group was to distinguish between non-homeomorphic topological spaces. So my question is, does fundamental ...
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0answers
25 views

Covering map, end points of two paths equal criterion

I was reading a proof of the following lemma: $\textbf{Lemma}$: Let $p\colon X \to B$ be a covering map and let $\gamma, \gamma'$ be two paths in $X$ beginning at $x_0$. Let $u=p\gamma$ and $u'=p\...
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2answers
56 views

Is there a retraction of $S^1\vee S^2$ onto $S^1$?

I am currently working through a problem that requires me to prove that $\pi_1(S^1\vee S^2)\simeq\mathbb{Z}$ directly by showing that the homomorphism induced by the inclusion of $S^1$ on $\pi_1$ is a ...
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1answer
43 views

Normal bundle of an embedding of a parallelizable manifold into a parallelizable manifold

This question is motivated by an observation of Milnor. Theorem: Let $M^m$, $N^n$ be parallelizable smooth manifolds, and $i:M\to N$ an embedding. If $n>2m$, the normal bundle is trivial. The ...
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1answer
28 views

Existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure

In deriving the existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure Ralph Cohen in his notes on the topology of fiber bundle (pp.169) uses the following diagram $\require{AMScd}$ \...
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1answer
19 views

Explicit formulation of hermitian form and corresponding alternating form

I can't understand the following basic thing: Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\...
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1answer
23 views

Properly discontinuous group actions - Hausdorffness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
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1answer
63 views

Hatcher's exercise 1.2.22 on the Wirtinger presentation

Here exercise 1.2.22 is recalled, but the asker seems to know how to solve it assuming the "geometry is valid". I however, do not know to use the van Kampen theorem in order to find the relations $...
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0answers
51 views

Topology of CW-complex and attaching map

I think I must have a fundamental misconception in place right now in my mind. When defining a CW-complex, we use inductively continuous maps from $f_{\partial \sigma} :S^n \to K^{(n)}$. We then ...
4
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2answers
65 views

Why is $H_5(K(\Bbb Z_n,4))$ finite?

I want to see that the cohomology $H^i(K(\Bbb Z_n,4); \Bbb Z)$ starts with $\Bbb Z_n$ in degree 5. How do we know that $\operatorname{hom}(H_5(K(\Bbb Z_n,4);\Bbb Z), \Bbb Z)$ is zero? I.e. why is $H_5(...
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0answers
57 views

Local isometry between non-positively curved cube complexes

Let $X$, $Y$ be non-positively curved cube complexes and suppose there is a local isometry $X \to Y$. If $Y$ is special, then so is $X$. This is Exercise 4.32 in M. Sageev's notes "CAT(0) Cube ...
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1answer
45 views

I want to prove that $B=S^2 \cap \{(x_1, x_2, x_3) : x_i \geq 0 \}$ is homeomorphic to the disk $B^2= \{ x \in \mathbb{R^2} : ||x|| \leq 1 \}$

I claim that the map $f:B \to B^2$ defined by $f(x_1, x_2, x_3)=(x_1, x_2, 0)$ is a continious, bijection with $g: B^2 \to B$, $g(x_1, x_2)=(x_1, x_2, \sqrt{1-(x_1^2+x_2^2)})$ it's inverse. So i ...
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0answers
19 views

Pictures and illustrations of attaching a 2-cell to a wedge of circles?

I am having a lot of trouble visualizing how to attach a disk "along a word". Where can I find some pictures and illustrations of this procedure?
3
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2answers
51 views

Does topological degree generalize to maps that aren't between closed connected orientable manifolds?

From what I gather, the degree of a map originally arose in the context of studying maps $f:S^n\rightarrow S^n$. Since $H_n(S^n)\cong \mathbb{Z}$, the induced map $f_\star$ has the form $x\mapsto kx$ ...
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0answers
58 views

Looking for a right book for Algebraic Topology - is Dieck's textbook a good choice?

I self-study Algebraic Topology. I use Hatcher's textbook Algebraic Topology and soon I'm going to end reading Chapter 3. I know that there is one more chapter about homotopy theory but I'd like to ...
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0answers
65 views

Covering map of classifying space

We know that for any $m\in\mathbb{N}$ the map $p_m:S^1\to S^1$ is an $m$-sheeted covering of $S^1$. Suppose that $BG$ is the classifying space of an arbitrary group $G$. Does there exist such a map $...
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2answers
71 views

Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
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1answer
26 views

What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ...
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0answers
25 views

Covering space, second-countable

Let $p:Y\to X$ be a covering space and $p^{-1}(x)$ countable for every $x\in X$ Show, that if $X$ is second-countable, so is $Y$ Hello, I am a little bit stuck with this question, because I do ...
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0answers
39 views

What does the operator “B” applied to an Eilenberg-MacLane space mean?

Let $K(m, \mathbb{Z})$ be the Eilenberg-MacLane space. I've read that $BK(m, \mathbb{Z}) = K(m+1, \mathbb{Z})$, but I'm trying to understand this. I'm familiar with the bar construction $BG$ for a ...
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0answers
108 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
3
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1answer
61 views

Automorphisms action on $\mathbb C$

If we know that the $$Aut \ \mathbb C=\{z\mapsto az+b:a\in \mathbb C^*,b\in \mathbb C\}$$ What are the subgroups of $Aut \ \mathbb C$ that act withiut fixed points and properly discontinuously on $\...
2
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1answer
70 views

There is no antipode-preserving map $f:S^{n+1}\rightarrow S^n$

I am trying to prove that there is no antipode-preserving map $f:S^{n+1}\rightarrow S^n$. Here is what I have: Suppose there is an antipode-preserving map $f:S^{n+1}\rightarrow S^n$. If we restrict ...
2
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1answer
49 views

how to represent a relative cohomology class

Let $X$ be a topological space, and $A \subseteq X$ a subspace. How to think about an element $u \in H^n(X, A)$? Is the following correct? $u$ can be represented by a function $U$ taking an $n$-cell $\...
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0answers
7 views

reference for simplicial complexes and triangulation

I am trying to understand simplicial complexes and triangulations, am following Basic Algebraic topology by Prof. Anant R Shastri and not very comfortable with the way it has been explained. Suggest ...
3
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1answer
35 views

Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
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33 views

Prove of homotopy equivalence using differential equation

I have to prove that $R^3 \setminus L_1,...,L_n $, where $L_1,..,L_n$ are non intersecting lines, is homotopy equivalent to a wedge sum of $n$ circles. So once I've managed to show that it is possible ...
3
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1answer
69 views

First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
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1answer
13 views

n-cubes vs simplices when defining chains

When defining chains, the standard definition is formal linear combinations of n-simplices. However in Calculus on Manifolds by Spivak, he defines chains as formal linear combinations of n-cubes. I ...
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1answer
78 views

Does any technical definition of embedding accept a “non-injective” function as opposed to only “injective”?

Embedding is defined to be a one-to-one structure preserving mapping. My question is if the one-to-one condition is really critical. Like if linear mappings from high-dimensional space to low-...
3
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3answers
105 views

Why bother showing $S^{1}$ covers itself?

I've just been introduced to covering spaces, and one of the examples I've been shown is that $p: S^{1} \to S^{1}$, $p(z)=z^{n}$ is a covering map for every $n$. My question is: why would you care? ...
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1answer
24 views

Map of degree two from $S^2$ to the torus $T^2$. [duplicate]

Prove that there is no map of degree two from $S^2$ to the torus $T^2$. I'm struggling with this problem. I've tried lifting the map to the covering space but I'm not sure what to do from there. I ...
0
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1answer
42 views

Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don't come along with this, ...
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1answer
35 views

Proving homotopy equivalence of a torus with points removed

Suppose I'd like to show that a Torus with $n$ points removed is homotopy equivalent to a wedge sum of $n+1$ circles. I depict it in a usual way - as a rectangle with $n$ points removed inside. Now it ...
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0answers
37 views

Hatcher Covering Spaces Ex. 11 & 31 and Surjectivity of the Covering Map

I am confused by the statements of a couple of the exercises in Section 1.3 of Hatcher. I think they need additional hypotheses that are not reflected in Hatcher's errata. Exercise 11: Construct ...
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0answers
36 views

geometric realization of a simplicial complex

Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard ...
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0answers
45 views

singular homology of a simplicial complex [duplicate]

On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the ...
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1answer
44 views

A deformation retract that is not a strong deformation retract

In Lee's Introduction to Topological Manifolds, problem 7-12 asks to show that $\{(1,0)\}$ is a deformation retract, but not a strong deformation retract of the subspace of the plane $$ X = \bigcup_{...
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1answer
62 views

Natural isomorphism $\tilde H_i(X) \xrightarrow{\cong} \tilde H_{i+1}(\Sigma X)$ where $\Sigma X$ is the suspension of $X$.

Define $\Sigma X$ to be the quotient space of $[-1,1]\times X$ obtained by identifying ${0}\times X$ and ${1}\times X$ to two points respectively. For any homology theory (satisfying Eilenberg-...
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1answer
27 views

Show that Borsuk lemma need not hold if $f$ is not injective

The following lemma is called Borsuk lemma which can be found in Munkres' topology (Lemma 62.2). (Borsuk lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let $f:A\to S^2\...
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1answer
24 views

A question about inherited orientation of simplices

I have the 2-manifold $[v_2,v_0,v_3]$. My books says that on removing $v_3$, the orientation of the face that we end up with is $[v_0,v_2]$. I don't understand how this happens. What is the ...
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0answers
18 views

A question about orienting a simplicial complex.

A complex may be oriented by assigning, in a completely arbitrary fashion, an orientation to each of its simplices I've always been confused about this point. Say we take a tetrahedron (a $3$-...
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0answers
39 views

What are some examples of cohomology theories without a corresponding classifying space?

The general nonsense of cohomology theories is that each one "should" be presented by a classifying space, so that maps into this space give the cohomology (before passing to connected components). ...
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1answer
16 views

Edge identification implies vertex identification (but not vice versa)?

This may be an obvious question, but just to ask it to be sure. Q1) When we identify edges, is it automatically assumed that we identify the vertices as well? E.g: We have a 2-simplex with vertices $...
1
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1answer
28 views

Is a $\Omega$-Spectrum a connective one?

I can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback Let $X$ be a $\Omega$-spectrum (of ...
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2answers
146 views

Is there a subject in mathematics like topological Algebra?

I would consider myself an algebraic topologist and there is a lot of influence from algebra into topology and without this input from the algebraic site I would say that a lot of topological theorems ...