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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1answer
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$\pi_1(S^n) = 0$ 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 ...
1
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0answers
59 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
69 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
46 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 ...
1
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1answer
50 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
50 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
45 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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2answers
80 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
36 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
57 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
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2answers
59 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
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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
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1answer
73 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
124 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
80 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 ...
0
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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
79 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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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
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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
73 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 ...
3
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1answer
52 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
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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 ...
3
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0answers
50 views

Isomorphism in integral Cohomology gives isomorphism in rational cohomology

I was asking myself the question, if a map $f\colon X \to Y$ between CW complexes gives an isomorphism between $H^*(X,\mathbb{Z})$ and $H^*(Y,\mathbb{Z})$ does it already give an isomorphism between ...
1
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2answers
51 views

Compact cohomology group of connected n-dimensional connected oriented manifold

I know how to show $H_c^n(M)\simeq\mathbb{R}$, where M is a oriented connected n-dimensional manifold, by showing the integration map is isomorphism. However, I found in the book that this is a ...
0
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0answers
16 views

Structure group reduction criterion in terms of classifying map lifting

I am looking for a proof of the following theorem which is given as an exercise in Ralph Cohen's notes on the topology of fiber bundles pp.74. But in view of its importance to the later chapters I ...
2
votes
1answer
52 views

Homeomorphism between N-disk and N-Projective Plane

I've just showed that: $D^n$, quotiented with this equivalence relation: $x\sim y \iff x=-y \text{ and } x,y\in\partial D^n$ (i.e. the antipodal points on the boundary of $D^n$ are identified) is ...
0
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2answers
31 views

Inverse mapping of a contractible space

So, let us be given a nonconstant continuous function $f:X \to Y$ and $B\subset Y$ is contractible. Now, I wonder if $f^{-1}(B)$ is contractible. I need this to solve a problem correctly(if you can ...
1
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1answer
50 views

$SO(3)$ homeomorphic to $\mathbb{R}P^3$

I'm doing some topological base-exercises, but I can't come up with this problem (That I suppose should be quite trivial): $SO(3)$ is homeomorphic to $\mathbb{R}P^3$. Any hints? thank you in Advance!...
3
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1answer
38 views

Proof of :$H^0(E;\pi_0E)\cong \hom_{\pi_0}(H_0(E;E_0);\pi_0E)$ for $E$ a multiplicative spectrum.

Let $E$ be a multiplicative spectrum, connective, and assume $\pi_0E$ is cyclic. I want to prove that $$H^0(E;\pi_0E)\cong \hom_{\pi_0E}(H_0(E;\pi_0E);\pi_0E)$$ Recall that $H^0(E;\pi_0E):= [E; K(\...
0
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1answer
54 views

Coefficients of homology

I am wondering why people use different coefficients when defining homology of simplicial complex, like homology over $R$, $Z$, $Z/2$, etc? Is one better then the other and why? Moreover, which one(s)...
3
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0answers
22 views

Complexes $K$, $L$, imply $|K| \cap |L|$ is polyhedron.

I am using Armstrong's topology text, and have been really stumped on what this question is asking. It says If $K$ and $L$ are complexes in $\mathbb{E}^n$, show that $|K| \cap |L|$ is a ...
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0answers
9 views

Iff conditions for acyclic, free, positive chain complexes with augmentation

I have some doubts about the formulation of the following lemma (from Ferrario, Piccinini - Simplicial structures in topology) and its proof. (II.3.8, page 72) Lemma. Let $(C,\partial) $ be a ...
0
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1answer
26 views

Why is the quotient map $G\rightarrow G/T$ a fibration?

I have just learnt about fibrations and I saw somewhere the following. Given a compact lie group $G$, one can consider a maximal torus $T$ of $G$ then the quotient map $G\rightarrow G/T$ is a ...
2
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0answers
19 views

Pairing on the AHSS induced by cap product: why does it exists

This is my setting: Let $\xi \colon X \to B$ be a vector bundle. Let $E$ be a ring spectrum. Suppose given a natural cap product $$ \frown \colon E^s(X,X\setminus B)\otimes E_{s+t}(X,X\setminus B)...
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0answers
42 views

Weibel 5.1.1 Exercise

I know this topic is already dealt on Total complex homology exact sequence, But have a question on the answer. The answer says that $$H_{p + q}(T) \cong \frac{\{(a,b) | d^v_{p-1,q+1}(a)+d^h_{p,q}...
3
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0answers
31 views

Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
0
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2answers
85 views

Reference for Algebraic Topology

I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should ...
0
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1answer
42 views

What is the topological degree of the constant map?

What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?
3
votes
2answers
59 views

Why $h$ has zero topological degree?

I am trying to prove that $f,g : M^n \to S^n$, both $C^1$ (indeed just $C^0$ is enough) with the same topological degree are homotopic. I saw on a book that the trick is as follows: Take $W = M\...