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

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0
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Hatcher 2.1.10…

Hatcher asked a question in chapter 2 (a) Show the quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic with ...
3
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3answers
84 views

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map?

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map? I get the picture that if the identity map $id$ is homotopic to the constant map then as the circle transforms ...
0
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2answers
58 views

Group with topology which is not topological group

What will example of a group $G$ with topology $\tau$ such that $f: G \to G$ such that $f(x) =x^{-1}$ and $g: G \times G \to G$ such that $g((x,y)) = x * y$ (where $*$ is binary operation on $G$) both ...
12
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2answers
124 views

Existence of submersions from spheres into spheres

I would like to know if there exist submersions $f\colon \mathbb{S}^4\to \mathbb{S}^2$ and $g\colon\mathbb{S}^6\to \mathbb{S}^2.$ It is simply a question of curiosity. Any suggestion or ideas are ...
2
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2answers
1k views

What is a good Algebraic topology reference text? [duplicate]

Possible Duplicate: Learning Roadmap for Algebraic Topology The title of the question already says it all but I would like to add that I would really like the book to be about more ...
4
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0answers
339 views

suggestion for video lectures on algebraic topology

can anyone suggest me any good video lecture series for algebraic topology other than N.J.Wildberger videos. If it is equivalent to Munkres topology (algebraic topology section) it should be great. ...
10
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2answers
372 views

Ehresmann Connection of the tangential bundle & Chern classes

I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking. The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
5
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1answer
111 views
+300

Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
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1answer
36 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
2
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0answers
57 views
+50

The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don't fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: ...
2
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1answer
51 views

Eilenberg–Maclane space: nonvanishing cohomology

I would like to prove (self study) that $H^{np}(K(\mathbb{Z},n),\mathbb{Z})\neq 0$, where $n$ is even , $p\geq1$ and $K(\mathbb{Z},n)$ is the Eilenberg–Maclane space. I used the adjunction between $[ ...
2
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0answers
31 views

Brouwer fixed-point theorem on non-convex sets [duplicate]

I read about the Brouwer fixed-point theorem in Wikipedia, and got confused about whether it holds when the domain of the function is non convex. On one hand, convexity is explicitly mentioned as a ...
3
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1answer
72 views

Counterexamples to Brouwer’s fixed point theorem

Brouwer’s fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least ones fixed point. If we replace the convex condition with let's say ...
5
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2answers
98 views
+50

Concatenating countably many homotopies

On page 15 of Hatcher's Algebraic Topology, he discusses constructing a homotopy $X \times I \to X \times \{0\} \cup A \times I$, where $(X,A)$ is a CW pair. He does so by concatenating homotopies ...
2
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1answer
36 views

$\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes

See the related question here. This is the second part of question 4-C in Milnor and Stasheff's book on characteristic classes. In the solution to the first part, we rely on the fact that having a ...
2
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0answers
92 views

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of ...
4
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1answer
91 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
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0answers
45 views
2
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3answers
155 views

Winding maps of spheres?

I know that the second homotopy group of $S^2$ is $\mathbb{Z}$. I also know that a simple representative of the class of maps that generates this group is the identity map on the sphere, ...
1
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1answer
74 views

Homotopy lifting property of $\mathbb{R} \to S^1$ in Hatcher

I am reading Hatcher's proof of the homotopy lifting property of the covering map $p: \mathbb{R}\to S^1$. Starting with a homotopy $F: Y \times I \to S^1$ and a map $\tilde{F}:Y \times \{0\} \to ...
0
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1answer
37 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
6
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1answer
57 views

When does covering preserve rational cohomology?

Let $X$ and $Y$ be compact manifolds. $p:X \rightarrow Y$ is a covering. Generally it is not true that $$ H^* (X , \mathbb{Q} ) = H^* (Y , \mathbb{Q} )$$ For instance, if $Y$ is a sphere with $y$ ...
5
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2answers
58 views

Prob. 5 (a) in Supplementary Exercises in Munkres' TOPOLOGY, 2nd ed: How to show that this map is a homeomorphism?

Let $G$ be a topplogical group, and let $H$ be a subgroup of $G$. Let $G / H $ denote the collection of all left cosets of $H$ in $G$, and let $a$ be a fixed element of $G$. Let the map $f \colon G / ...
7
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1answer
81 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
1
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0answers
40 views

Projection of a covering map over product set.

Let $p,q$ be a covering maps, $p:\tilde X \rightarrow X$ and $q:\tilde Y \rightarrow X$ and let $Z=\lbrace(\tilde x, \tilde y)| p(\tilde x)=q(\tilde y) \rbrace$, I want to proff that $f:\tilde ...
3
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2answers
114 views

First book on algebraic topology

Is there a good book on introduction to Algebraic Topology which is self-contained and does not assume any background in topology, only standard calculus and linear algebra courses?
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0answers
36 views

Cone of projective space?

Is the cone of $\mathbb{C}\mathbb{P}^2$ a familiar topological space? What about $\mathbb{C}\mathbb{P}^3$? I'm having a lot of trouble visualizing it. I just learned the notion of the cone of a ...
0
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0answers
27 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
6
votes
1answer
77 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
5
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1answer
46 views

Spectral sequence for computing the homotopy fixed points in unstable equivariant homotopy theory

I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question. Let $G$ be a topological group and $X$ be a $G$-space (for a nice notion of "space"). He defines ...
2
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1answer
57 views

Showing two spaces are homotopy equivalent

So I understand the basics about homotopy, I know a punctured disk or $\mathbb{R}- \{ 0 \}$ are homotopy equivalent to $\mathbb{S}^1$. This can be shown using the deformation retract ...
0
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1answer
74 views

Klein bottle and Real Projective plane

How to determine the triangulation of these two objects? can we use the above to compute Fundamental Group of Klein bottle and Real Projective plane? I can use the van kamen theorem to prove one is ...
1
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2answers
59 views

Splitting of Singular Homologies

In Singular homology, let $C_n(X)$ be the free abelian group generated by all the $n$-siimplices of the topological space $X$. Let $U$ be a subspace of $X$, then we have a spliting sequence ...
1
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0answers
60 views

Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...
5
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2answers
47 views

Even number of points in the preimage of a regular value of a map $f:M^n \to \mathbb{R}P^n$

Let $M^n$ be a closed manifold (I think we can also assume it is connected, though it isn't explicitly stated). Let $f:M^n \to \mathbb{R}P^n$ be a smooth map, and let be $a \in \mathbb{R}P^n$ be a ...
5
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1answer
45 views

If the top Stiefel-Whitney class of a compact manifold is nonzer0, must there be another non-vanishing Stiefel-Whitney class?

I was trying to collect some examples of Stiefel-Whitney class computations, just to make myself more familiar with them. It seems from my (relatively short list) that if the top Stiefel-Whitney ...
2
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1answer
37 views

$H_n(S^n,A)$ is not trivial

Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be an ordinary homology theory with values in the category of $R$-modules. Let $A\subset S^n$ be a proper subset. Then $H_n(S^n, A)$ is not ...
2
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2answers
307 views

Euler class of tangent bundle of the sphere

I am working through Milnor's Characteristic classes and am currently working problems on the topic of oriented bundles and euler class. I am having trouble computing the euler class of the tangent ...
4
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1answer
1k views

Homology of wedge sum is the direct sum of homologies

I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a ...
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2answers
66 views

Why does $\overline{\alpha * \beta}=\bar{\beta} * \bar{\alpha}$

I'm working on this question from Munkres' topology: Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$; let $\beta$ be a path in $X$ from $x_1$ to $x_2$. Show that if $\gamma = \alpha * \beta$ , then ...
18
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6answers
2k views

What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? ...
4
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1answer
59 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
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1answer
223 views

What does $b_i\mid b_{i+1}$ mean in this context?

In the computational topology literature, the reduction algorithm for computing the Smith normal form of a boundary matrix uses the notation $b_j > 1 \: \text{ and }\: b_j\mid b_{j+1}$ in the ...
3
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1answer
315 views

Bing's House and homotopies

Muestra que si f es la función que encaja a $S^{1}$ en la circunferencia que rodea al cilindro mayor, por la mitad, de la casa de Bing, entonces f es homotópica a una constante. Show that if $\,f\,$ ...
6
votes
3answers
128 views

$S^m * S^n \approx S^{m+n+1}$

I'm interested in showing that $S^m * S^n \approx S^{m+n+1} $, as discussed in exercise 0.18 of Hatcher's Algebraic Topology. One way to show it would be to show that $X * Y \approx \Sigma(X \wedge ...
3
votes
1answer
34 views

Inclusions of CW-complexes are cofibrations.

Has the inclusion from the $ (n - 1) $-sphere in the $ n $-disc the left lifting property for all acyclic Serre fibrations? I am looking for a reference for this proposition, or alternatively, for an ...
0
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0answers
37 views

Proving two spaces are homotopy equivalent

We are given a topologic space X, defined as: $$X= \mathbb{S}^2 \cup \mathbb{D}_2 \cup \mathbb{I} \subset \mathbb{R}^3$$ Where $$\mathbb{S}^1=\{ (x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1 \} $$ ...
2
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1answer
49 views

Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
2
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1answer
46 views

Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
2
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1answer
254 views

Two deformation retractions (onto $A$) are homotopic (rel $A$).

This is a question from Hatcher's Algebraic Topology (Chapter 0, Question 13): 13. Show that any two deformation retractions $r^0_t$ and $r^1_t$ of a space $X$ onto a subspace $A$ can be joined by ...