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

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1
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2answers
324 views

Question about two simple problems on covering spaces

Here are two problems that look trivial, but I could not prove. i) If $p:E \to B$ and $j:B \to Z$ are covering maps, and $j$ is such that the preimages of points are finite sets, then the composite ...
2
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1answer
120 views

starting with paths and the fundamental group

I have two problems, I´m a little complicated with the problem , I know that it´s easy but I need just a little help. Are two problems i) Suppose that the identity map $ i:X \to X $ is homotopic to ...
11
votes
2answers
472 views

Decomposing the sphere as a product

It seems obvious that, if the sphere $S^n$ is homeomorphic to a product $X\times Y$ of topological spaces, then either $X$ or $Y$ is a point. How can one prove that?
6
votes
1answer
162 views

Why $T\mathbb{S}^{2}\otimes T\mathbb{S}^{2}$ is trivial?

I encountered this problem in an homework problem set of algebraic topology. Naturally I thought about Bott Periodicity which implies $K(\mathbb{S}^{2})\cong \mathbb{Z}$. But I am beware that we are ...
2
votes
2answers
163 views

Relationship between the $2$-plane bundles over $S^2$ and $\mathbb{Z}$

I want to follow up on this answer by asking a few more questions (posting directly on the question didn't seem to "bump" the thread). I was trying to read the referenced text (Husemoller's Fiber ...
1
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1answer
201 views

Algebraic Topology in Simple Terms

I just wanted to clarify a few basic concepts in algebraic topology. Suppose one space is my room ($\text{Room} \ A$). Suppose the other space is another room in my house ($\text{Room} \ B$). So ...
6
votes
1answer
314 views

What tells rational cohomology about integral cohomology?

Say we have a finite CW complex with cells only in even degrees. For example a $\mathbb {CP}^n$ or a complex flag variety. If we know the rational cohomology ring, does it also determine the integral ...
4
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3answers
1k views

Another Question in Hatcher

First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again. The problem is 1.3.15. It reads ...
1
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1answer
281 views

How does this proof show $h_*$ (homomorphism induced by $h\colon (X,a)\to (Y,b)$) is an isomorphism?

Claim: if $h\colon(X,a)\to(Y,b)$ is a homeomorphism of $X$ with $Y$, then $h_*\colon \pi_1(X,a)\to \pi_1(Y,b)$ is an isomorphism. where $\pi_1$ refers to the fundamental group and $h_*$ is the ...
2
votes
1answer
96 views

Shellable and Graphs

Suppose we have a graph $G$ of order $n$. Also suppose that we form the coloring complex $S(G)$ of $G$. What does it mean when we say that $S(G)$ is shellable?
2
votes
1answer
772 views

Euler characteristic of sphere with a hole

The topologically invariant Euler characteristic of a 2-surface is given by $\chi=\frac{1}{4\pi}\int\sqrt{g}\mathcal{R}$ (where $\mathcal{R}$ is the scalar curvature) and is equivalent to $\chi=2-2g - ...
5
votes
0answers
109 views

Fundamental group of the complement of a linear subspace

Let $m<n-1$ be two positive integers. Consider $\mathbb{R}^m$ as a subspace of $\mathbb{R}^n$ via $\mathbb{R}^m\times \{(0,0,...0)\}$. Any suggestions on how to compute ...
1
vote
1answer
229 views

Double of a manifold

Let $M$ be a connected $n$-manifold with a non-empty boundary. The double of it is given by $$ D(M) = M\,\,\,\cup_f\,\,\, M $$ where $f:\partial M\to\partial M$ is an identity map. I have to show ...
7
votes
2answers
734 views

The image of simply-connected domain

If $\Omega$ is a simply-connected domain in $\mathbb R^n$ and $f$ is a injective continuous map from $\Omega$ to $\mathbb R^n$, then is it necessary that $f(\Omega)$ a simply-connected domain?
1
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1answer
80 views

Decomposition of simplicial G-sets as a colimits of its simplicial G-subsets

Every simplicial set is the colimit of its finite simplicial subsets. Suppose $G$ be a finite discrete set. Is every simplicial $G$-set a colimit of its finite simplicial $G$-subsets? I'm ...
2
votes
1answer
124 views

Adjointness of the corresponding simplicial functors associated to an adjoint pair between categories

Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. ...
4
votes
1answer
563 views

Does the homology, homotopy, and geometric realization functors of a simplicial group preserve colimits?

Given a based simplicial group, you can find its reduced homology with coefficients in a field, homotopy, and geometric realization. These are functors. If I have a free product of based simplicial ...
4
votes
2answers
449 views

Good Reference for Spanier-Whitehead duality?

Does anyone know of a good book that explains Spanier-Whitehead duality (other than Adams)? Thanks Jon
34
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2answers
1k views

How much rigour is necessary?

I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in ...
6
votes
0answers
419 views

Homotopy extension property vs. good pairs

I'm taking a course that uses the book "algebraic topology" by Allen Hatcher. I this book there are two different ways in which a pair (X,A) of a topological space X and a subspace A can be nice: They ...
1
vote
0answers
58 views

General convergence criterion for a spectral sequence associated to a filtration of simplicial groups

Let $\cdots\subset F_r \subset \cdots \subset F_1 \subset F_0=G$ be a filtration of simplicial groups. If $\bigcap F_r$ is trivial, does the associated spectral sequence converge? If so, what does ...
1
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1answer
145 views

Question on transitivity of proof that path homotopy induces an equivalence class (munkres topology)

(If you have the text)pg 324 of Munkres topology. If $F$ and $F'$ are path homotopies between $f$ and $f'$ & $f'$ and $f''$, respectively, Munkres defines a path homotopy $$ ...
2
votes
0answers
58 views

ruling out non Pseudo-Anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
6
votes
1answer
256 views

Are all vector bundles “flat vector bundles”?

This concept appears in Bott&Tu's GTM82. A flat vector bundle is one who has a particular trivialization with locally constant transition functions. Then my question is whether every vector bundle ...
1
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1answer
136 views

Number of components of complement to a reducible real algebraic hypersurface

Let $X_1,\ldots X_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x_1,\ldots, x_k$ connected components, respectively. Let $G_1,\ldots, G_l$ be their ...
3
votes
1answer
69 views

The cofibre of a type $n$ $p$-local CW complex has type $n+1$

I am trying to prove this relatively simple statement from 'Nilpotence and periodicity in Stable Homotopy Theory' Suppose $X$ as in the periodicity theorem has type $n$. Then the cofibre of the ...
2
votes
0answers
94 views

Proof of excision

The following link is the proof of excision from Rotman's Algebraic Topology. Please see the the third line of the page 118. I am wondering why the equation is held. Note that it is composition of q ...
4
votes
1answer
176 views

$G/H$ with $H$ contractible a fibre bundle?

Suppose $G$ is a topological group and $H\leq G$ a normal/closed subgroup of $G$. If $H$ is contractible, does the quotient map $p: G\rightarrow G/H$ form a fibre bundle? Is there a more general ...
4
votes
2answers
214 views

Is the suspension of a $\pi_n$ isomorphism a $\pi_{n+1}$ isomorphism?

Let $$f_*:\pi_n(A)\to\pi_n(X)$$ be an isomorphism induced by a map $f:A\to X$ of based topological spaces. Does the suspension $\Sigma f$ induce an isomorphism $$(\Sigma_f)_*:\pi_{n+1}(\Sigma ...
6
votes
1answer
123 views

Difference between X/A and G/H

I am primarily a student of physics and am trying to self-learn some algebraic topology. I am having some difficulty understanding the differences between the constructions of $(X,A)$ (Pair of ...
9
votes
2answers
248 views

Topological rings which are manifolds

Is the following statement true: "Every smooth manifold $M$, which is a ring in the category of manifolds, must be diffeomorphic to $\mathbb{R}^n$."? (Actually, homeomorphic would suffice.) I assume ...
3
votes
1answer
203 views

Does the Sorgenfrey Line have a group operation compatible with its order topology?

The title is the question, but let me explain. Let $\mathbb{L}$ denote the Sorgenfrey line. I and a friend were trying to develop some of the properties of the sorgenfrey line. (if it's metrizable, ...
4
votes
2answers
360 views

$S^n \backslash S^m $ homotopy equivalent to $ S^{n-m-1} $

Consider $S^m$ embedded in $S^n$ $ (m < n ) $ as the subspace $ \left\{ (x_1, x_2, ..., x_{m+1}, 0,...0) | \sum x_i^2 = 1 \right\} $. Show that $ S^n \backslash S^m $ is homotopy equivalent to $ ...
6
votes
1answer
208 views

Why does Frank Adams demand a finite CW-complex?

On page 145 of J.F. Adams' "Stable Homotopy and Generalised Homology", there is a proposition: Let $E$ be the suspension spectrum of a finite CW-complex $K$, and $F$ and spectrum (of topological ...
0
votes
1answer
130 views

Homotopy equivalence definition

I've been given the question: Let $ f : X \to Y $ be a continuous map, and suppose we are given (not necessarily equal) continuous maps $ g,h : Y \to X $ such that $ gf \simeq id_X $ and $ fh \simeq ...
1
vote
1answer
325 views

Which of the letters of the alphabet are contractible?

I've been asked which of A,B,C...,Z are contractible. Intuitively, I can see that all but A, B, D, O, P, Q and R are contractible. These are all (except B) homotopy equivalent to a circle, and B ...
3
votes
0answers
189 views

The fundamental group of the mapping torus is doubly degenerate

Consider an hyperbolic compact surface $S$ (hence with genus $>1$) and a Pseudo-Anosov diffeomorphism $\varphi\colon S\to S$. We call "mapping torus" the 3-manifold ...
6
votes
1answer
207 views

Obstructions to lifting a map for the Hopf fibration

This is a bit of an elementary question, but suppose $\pi: \mathbb{S}^3\to \mathbb{S}^2$ is the Hopf fibration, are there reasonably computable obstructions to when a map $f:M\to \mathbb{S}^2$ can be ...
3
votes
1answer
190 views

Basic property of homotopy

Suppose $ f : X \to Y$ and $g,h : Y \to Z$ are continuous, with $g \simeq h$. Prove that $ gf \simeq hf $. My attempt: Suppose $ L(x,t) $ gives a homotopy from $g$ to $h$, i.e. $ L(x,t) : Y \times ...
1
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0answers
443 views

Examples of fundamental group of mapping torus

The original question was askes here. I donot know how to apply or compute any example. I think a specified explanation will be helpful. Let $M=\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and ...
0
votes
1answer
459 views

Suspension of a map

Suppose I have a surjective map, say $f$, between two spheres (of dimension $n+1 \geq 2$) such that it takes the closed upper hemisphere to itself and the closed lower hemisphere to itself. Now, I get ...
3
votes
4answers
2k views

Antipodal map on $ S^n $ homotopic to identity map if $n$ is odd

I want to prove that the antipodal map from $S^n$ to $S^n$ is homotopic to the identity map if $n$ is odd. (I know it's actually true if and only if) If I consider the map $ H(x,t) = (1-2t)x $, why ...
22
votes
3answers
1k views

Why algebraic topology is also called combinatorial topology?

I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...
2
votes
1answer
87 views

Homotopy of Spectra Maps Induced by Homotopy of Functions

So, I'm still working through Adams' lecture notes, and here's something that I haven't been able to immediately suss out: It is clear that for $F$ a spectrum, $K$ a finite CW-complex and $E$ its ...
4
votes
1answer
111 views

Subcategory of (co)fibrant objects of a model category

Maybe that's a very naive question. Is the subcategory of cofibrant and fibrant objects of a model category a model category itself (with the induced equivalences, cofibrations and fibrations)?
2
votes
0answers
199 views

Developing a counter-example in algebraic topology

Two maps $f,g$ into $Y$ are n-homotopic if, for every complex $K$ of dimension at most $n$ and for every map $\phi$ of $K$ into $X$ the compositions $f \phi,g\phi:K \to Y$ are homotopic. As a sort ...
9
votes
1answer
536 views

Question in Hatcher

Exercise 0.21 of Hatcher's Algebraic Topology reads: If $X$ is a connected Hausdorff space that is the union of a finite number of $2$-spheres, any two of which intersect in at most one point, ...
3
votes
1answer
111 views

Not homotopy equivalent to a 3-manifold w/ boundary

Let $X_g$ be the wedge of $g$ copies of the circle $S^1$ where $g>1$. Prove that $X_g \times X_g$ is not homotopy equivalent to a 3-manifold with boundary.
3
votes
2answers
394 views

Hausdorff condition for CW complexes

Consider the definition of CW complex from wikipedia. It is assumed that the space is Hausdorff. Are there problems if we drop this assumption? What is an example of a space satisfying all the CW ...
3
votes
0answers
175 views

Local contractibility of CW complex

I was trying to understand the notion of CW complex from wikipedia. The very first non-example is: $$\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2$$ This is not ...