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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28 views

Proof of the cellular boundary formula

I'm trying to understand the proof in Hatcher (p. 141) of the cellular boundary formula. Now there's one thing that Hatcher does several times in his book and that I don't understand very well: he ...
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0answers
54 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...
2
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1answer
40 views

Proving that a map is a weak homotopy equivalence

Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting diagrams where all the morphisms are serre fibrations $$\begin{...
6
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1answer
171 views

How to know if you are “tough enough” to study Algebraic Topology [closed]

I am graduating with a BA this summer and I am very interested in topology. I admit it, I never went that deep into topology and all I know is about point-set topology (metric spaces etc.) but from ...
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1answer
45 views

Projective space, $S^n$

We observe the projective space $\mathbb{R}P^n$ for $n>1$. Let $e\in S^n$ be random. a) The quotient map $p:S^n\to\mathbb{R}P^n$ is an overlapping and $U_i:=\{p(x):x\in S^n, x_i\neq 0\}\...
6
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3answers
94 views

Intuitive reason why the Euler characteristic is an alternating sum?

The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-...
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1answer
44 views

A modern approach to homotopy theory in $\mathbf{SSet}$

I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. ...
2
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1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
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0answers
23 views

Let U be a simple connected open set in R×R. If C is a simple closed curve lying in U, then each bounded component of R×R - C also lies in U.

Suppose U = B(0,2), C is a simple closed curve lying in U, then theorem obviously true competition. Let U be a simple connected set in R×R?
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0answers
60 views

Borsuk lemma doesn't hold if f is not injective. [closed]

Give an example to show that the conclusion of the Borsuk lemma need not hold if the map is not injective. Statement of the aforementioned lemma: Let $A$ be a compact topological space and $f:...
5
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1answer
77 views

Why does Van Kampen Theorem fail for the Hawaiian earring space?

The Hawaiian earring space has notoriously complicated fundamental group, and is essentially not as simple as the wedge sum of countably many circles whose fundamental group is straightforwardly given ...
2
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1answer
49 views

Degree formula for smash product

Let $f : S^n \to S^n$ and $g : S^m \to S^m$ be two maps with degrees $d_f$ and $d_g$ respectively. These two map gives rise to a map $f \wedge g : S^{n+m} \to S^{n+m}$. My question is how the degree ...
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2answers
56 views

submanifold with same homology

Suppose $M$ is a manifold without boundary, and $N\subseteq M$ is any submanifold, possibly with boundary. If $H_*(N)\cong H_*(M)$, is it necessarily true that $N\cong M$?
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0answers
36 views

Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
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0answers
15 views

winding number, “function degree”

For every continuous function $f: S^1\to S^1$ with $f(1)=1$ we define the "function degree" $deg(f)$ as winding number of the path $f\circ w_1$ Show, that: a) $deg(f)=n\Leftrightarrow f\circ ...
2
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0answers
33 views

Proving that Emb($D^m,N$) is homotopy equivalent to $V_m(TN)$

I am reading online lecture notes by John Francis on h-principle. I want to prove that Emb($D^m,N$) is homotopy equivalent to $V_m(TN)$ where $V_m(TN)$ is stiefel bundle of the tangent bundle on $N$....
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1answer
50 views

$\pi_1(S^n) = 0$ for $n \geq 2$

I have few of questions for the following proof in hatcher's book. (1)Why f being continous imply that for each $s \in I$ has an open neighborhood $V_s$ in I mapped by f to some $A_{\alpha}$. (2)Why ...
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0answers
57 views

Generalized Euler Characteristic

I was asking myself what kind of generalizations of the euler characteristic are there? I've heard about the homotopy cardinality, but I was rather interested in a construction involving generalized ...
4
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0answers
67 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy.

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
2
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1answer
45 views

Cohomological AHSS for projective space $\mathbb{C}P^n$ ALWAYS collapses at the second page

While I was computing the cohomology ring $E^*(\mathbb{C}P^n)$ for $E$ an oriented ring spectrum, via AHSS I realised that orientation is not a necessary hypothesis in order to compute the cohomology ...
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1answer
48 views

Is the rank of this group always finite? Why?

Take $U$ an open subset of the plane. Consider $C_0(U)$ the free abelian group over the points of $U$, $C_1(U)$ the free abelian group over continuous paths (i.e. continuous maps $[0,1]\to U$), and $...
1
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1answer
27 views

Proving that a map is Null Homotopic

Suppose $X$ is a manifold of dimension $n$ and $f:Y \to Z$ is an $n-$connected map. Then I want to show that given any map from $g:X \to Y$, the composite map $f \circ g$ is nullhomotopic. Definition ...
2
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1answer
48 views

How to conceptually visualize the homotopy map?

I hope to be clear in my question, I've been meditating on the definition of Homotopy of two continuous maps and I've come to the following thought: This is the definition I'm adopting: let $f_0, f_1:...
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0answers
44 views

Relation between homology class and homology groups and betti numbers

I am reading about algebraic topology from various different sources. I found a lot of material on calculating homology groups using chain complexes and computing their betti numbers. I think I have ...
6
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1answer
71 views

Infinite sequence of distinct spaces, all with same homology

Using the following fact, we get infinitely many non-homotopic maps $f_k:S^{2n-1}\to S^n\vee S^n$. Fact: $\pi_{2n-1}(S^n\vee S^n)$ contains a $\Bbb Z$-summand. So we can consider the spaces $X_k=...
3
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1answer
46 views

If $(X, A)$ is a Good Pair then $i_*:H_n(X, A)\to H_n(X, V)$ is an Isomorphism

Definition. Let $A$ be a closed subspace of a topological space $X$. We say that $(X, A)$ is a good pair if there is a neighborhood $V$ of $A$ in $X$ which deformation retracts to $A$. In the proof ...
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1answer
33 views

homotopy groups of a pair and quotient

It is known that if $(X,A)$ is a good pair, for example a $CW$ pair, then $H_k(X,A)\simeq H_k(X/A)$ for every $k$. Is it true for homotopy groups of $CW$ pairs? If not, what is the counter-example?
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1answer
56 views

Singular homology groups of $S^5 - t^2$

Let $t^2 \subset S^5$ be a homeomorph of the two torus $T^2$. How can we compute the homology groups $H_* (S^5-t^2;\mathbb{Z})$? I know how to compute $H_* (S^5-s;\mathbb{Z})$ if $s$ is a homeomorph ...
2
votes
2answers
58 views

non-homotopic maps $\mathbb S^2\rightarrow \mathbb{P}_\mathbb{R}^2$

How can I find non-homotopic maps $\mathbb S^2 \rightarrow \mathbb{P}_\mathbb{R}^2$? I know that it is enought that the degree are different. But the canonical map given by the antipodal map has ...
0
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1answer
49 views

Covering space, product topology

Let $p_i: Y_i\to X_i$ with $i=1,2$ be covering spaces. Show, that $p_1\times p_2: Y_1\times Y_2\to X_1\times X_2$ is a covering space. Hello, I want to prove this statement. Therefore I want to ...
2
votes
2answers
71 views

Open ball does not have fixed point

How we can prove that the open ball in $R^n$ does not have fixed point property (by algebraic topology concepts)? I know $D^n$ -closed ball in $R^n$- has fixed point property by Brouwer's theorem, but ...
0
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1answer
56 views

Why is the following map from $S^1$ to $S^2$ null-homotopic

I am reading the following proof from hatcher. There is a certain point I don't understand. Why is the map given by $\eta : S^1 \to S^2$ null-homotopic in $S^2$?
1
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1answer
70 views

Whitney sum bundle vs. direct product bundle [closed]

The question is simple: are Whitney sum bundle and direct product bundle the same? When are they different? PS: I have realized my mistake: let $E_1 \to B$ and $E_2 \to B$ be two bundles, their ...
0
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0answers
29 views

Prove Thom isomorphism theorem using universal coefficient theorem

In Ralph Cohen's notes on the topology of fiber bundles pp.90, he claims that for the trivial bundle $p_{\xi}: X \times \mathbb{R}^n \to X$ Thom isomorphism follows from applying the universal ...
3
votes
1answer
70 views

What is the topology of an infinite cylinder?

Consider an infinitely long straw. This is a genus 1, orientable manifold. It is not closed because it is infinitely long. Is there a way I can describe the property that it is "partially closed" or ...
0
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1answer
45 views

The first Stiefel-Whitney class is zero if and only if the bundle is orientable

Ralph Cohen's notes on the topology of fiber bundles pp.84 (theorem 3.3) says that it follows immediately from the definition of the first Stiefel-Whitney class of real vector bundles (pp.83) \begin{...
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0answers
39 views

A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
2
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0answers
24 views

Classifying Unitaries on the Circle with K-Theory

Let $S^n$ be the $n$-sphere. I'm trying to understand the "meaning" of the $\mathbb{Z}$ factors in $$ K_0(C(S^{2n+1}))\cong\mathbb{Z}$$ and $$ K_0(C(S^{2n}))\cong\mathbb{Z}\oplus\mathbb{Z}$$ So $S^n$...
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2answers
122 views

Is $n$-dim manifold which is immersed in $\mathbb{R}^n$ diffeomorphic to a ball?

Let $M$ be an $n$-dimensional smooth compact oriented manifold with boundary which is immersed in $\mathbb{R}^n$ (codimension zero). Must $M$ be diffeomorphic to a ball with boundary (the closed unit ...
0
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2answers
72 views

Is there a non-trivial connex compact orientable topological manifold of Euler characteristic 1?

Is there a non-trivial connex compact orientable topological manifold of Euler characteristic $\chi = 1$? Remark: the point has $\chi = 1$, but it is trivial. The real projective plane has $\chi = ...
2
votes
1answer
44 views

Characteristic class invariant under bundle isomorphism

Let $c$ be a characteristic class for principal $G$-bundles and $p_1: E_1 \to X, p_2: E_2 \to X$ be isomorphic principal $G$-bundles, then $c(E_1) = c(E_2)$ Is this part of the defining naturality ...
5
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1answer
79 views

Basic Algebraic Topology puzzler

I've been watching Norman Wildberger's lectures on Algebraic Topology and one of his problems really got me stuck. The question is to show how a double-holed torus with a line of infinite length ...
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0answers
38 views

The complement $\mathbb{R^3}-A$ of a single circle $A$ deformation retracts onto the wedge of a Circle and a sphere. 2$

Here is my intuitive idea. We can pull all the points lying outside a sphere of some fixed radius containing the circle onto the sphere without disturbing the points inside the sphere. Now we have a ...
0
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1answer
78 views

Homotopy, Identity

Show that there is no homotopy between the identity and the function $f:S^1\to S^1$, $(x,y)\mapsto (x,-y)$ Hello, I have a problem with this task (the task got corrected), because I am not sure, ...
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votes
1answer
115 views

An application of Euler Characteristic to Tetrahedron Packing

The following is an application of Euler's equation to tetrahedron packing of any convex polyhedron. I related it to Euler formula; consequently, a third equation is obtained which is independent of ...
2
votes
1answer
51 views

Exact sequences of bundles and orientations

If we have an exact sequence of finite-dimensional vector spaces $$0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow0$$ then an orientation of any two induces an orientation of the third. I have ...
2
votes
1answer
71 views

The identification $G=\Omega^\infty \Sigma^\infty S_0$ of the stable group of self homotopy equivalences of spheres with the suspension spectrum

My topology professor told me in a discussion that the suspension spectrum $colim \Omega_n \Sigma_n S_0$ is the same as the monoid $G$ where $G=colim G_n$ where $G_n$ are self homotopy equivalences of ...
2
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0answers
46 views

Difference between product projections and split epis in $\mathbf{Top}$

I don't understand the excerpt below. The trivial bundles were defined as split epimorphisms, and since the ground category was additive with kernels (in fact abelian), they are the same as ...
0
votes
1answer
56 views

Relative homotopy

Show, that the functions $g: S^1\to S^2$, $(x,y)\mapsto (x,y,0)$ and $h: S^1\to S^2$, $(x,y)\mapsto (x,-y, 0)$ are relative homotopies to $(1,0)\in S^1$ Hello, I have a question to this task. I ...