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

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8
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1answer
443 views

The Mayer-Vietoris sequence

If $X$ is a space with a pair of subspaces $A, B \subset X$ such that $X$ is the union of the interiors of $A$ and $B$, then there is a long exact sequence of homology groups $\displaystyle \ldots\to ...
2
votes
1answer
450 views

exact homology sequence associated to a fibration

let $p:E\rightarrow B$ be a fibration with fiber $F$, ($E$ and $B$ are cw complexes). $B^k$ denotes the $k$-skeleton of $B$. 1) what does this sentence mean :"we denote the restriction of $E$ to ...
3
votes
0answers
519 views

Induced map in (co)homology in a map from torus to sphere

Let $f:T^2 \to S^2$ be the map that collapses the 1-skeleton $S^1 \vee S^1$ Compute $f_∗$,$f^*$. So start with the usual CW strucutre on the sphere (1 0-cell and 1 2-cell) and the torus (1 0-cell, 2 ...
0
votes
2answers
128 views

Continuous mapping $D^2 \to S^1$

It is a well known result in algebraic topology that there is no retraction of $D^2$ onto $S^1$. Does anyone know any continuous maps $D^2 \to S^1$ which are not constant?
2
votes
1answer
168 views

The quotient of a graph obtained from the universal cover of a double mapping cylinder

I'm having trouble with a problem from Hatcher's "Algebraic Topology" (Section 1.3, Problem 33). I'll provide a rough summary of the objects involved, but for a more precise description, have a look ...
10
votes
1answer
1k views

References for calculating cohomology rings

I am struggling to calculate homology rings. Even for a simple space such as the sphere, it is easy to calculate the cohomology, but I find it much harder to find the ring structure. (This link gives ...
9
votes
2answers
322 views

Etymology of the name “deck transformation”

What does the word "deck" mean in "deck transformation"? What's the idea behind this name?
3
votes
0answers
83 views

rationalization space

let $X$ be a topological space and $X_\mathbb Q$ its rationalization. 1) what is the rationalization of $X_\mathbb Q$, is it $X_\mathbb Q$ itself? 2) if $X$ is a CW complex, does that imply ...
2
votes
0answers
228 views

Naturality of the cross product

Apologies for the vague title, but I can't describe the following question well. The question is (as usual, from my book) Let $f:X \to X'$ and $g:Y \to Y'$ be continuous. If $\mbox{cls} ...
0
votes
1answer
221 views

How to prove that the intersection of two polyhedrons is still a polyhedron?

I've met a problem in M.A.Armstrong's Basic Topology. If $K$ and $L$ are complexes in $\mathbb{E}^n$, show that $\vert K\vert\cap\vert L\vert$ is a polyhedron. where $\vert K\vert$ and $\vert ...
11
votes
0answers
463 views

A Way to make the following “proof” of the Hairy Ball Theorem rigorous?

I plan on giving a talk soon to undergraduates and I'd like to talk about the hairy ball theorem during the talk. I was trying to think of some sort of visually intuitive proof of this fact. (I ...
14
votes
2answers
646 views

$X$ and $Y$ are homotopy equivalent $\Leftrightarrow$ $\exists Z:$ $X,Y$ are strong deformation retracts of $Z$

This question is very similar to this one, but the difference is that I'm asking for a strong deformation retraction. Notation/Definitions: (all maps are by definition continuous) A homotopy between ...
1
vote
1answer
528 views

Quotient space from the 8-sided polygonal region; Munkres

Let $X$ be the quotient space obtained from an 8-sided polygonal region $P$ by pasting its edges together according to the labelling scheme $acadbcd^{-1}d$. a) Check that all vertices of $P$ are ...
0
votes
1answer
109 views

attaching a cone

why $(X\cup CA)/CA=X/A$ and $(X\cup CA)/X=\Sigma A$ where $CA$ is the cone on $A$ and $\Sigma A$ is the suspension of $A$.
2
votes
1answer
999 views

Regular covering maps

This is a question from an old exam paper (Question 21G); regular covering maps were not lectured this year so I'm having a bit of trouble with it. (The material is not examinable this year, so this ...
10
votes
0answers
2k views

deformation retract and strong deformation retract

I am trying to gain some intuition about retracts, deformation retracts and strong deformation retracts (see http://en.wikipedia.org/wiki/Deformation_retract for definitions). We have that any strong ...
14
votes
1answer
400 views

What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these ...
1
vote
0answers
234 views

Applications of universal coefficient theorem

I'm not really sure what I want to ask here, which isn't a great start for a question, but nonetheless... I am wondering if there are some nice results that we can get from considering (co)homology ...
2
votes
1answer
192 views

Entangled circle in a solid torus (follow up)

I asked this question here. Can someone tell me if this is right: claim: There are no retractions $r:X \rightarrow A$ proof: (by contradiction) (i) If $f:X \rightarrow Y$ is a homotopy equivalence ...
1
vote
1answer
655 views

Lefschetz number equal to Euler Characteristic of Fixed Points

If $X$ is a finite simplicial complex and $f:X\rightarrow X$ is a simplicial homeomorphism, show that the Lefschetz number $\tau(f)$ equals the Euler characteristic of the set of fixed points of $f$. ...
2
votes
2answers
1k views

Quotient space of the torus with two points identified

Take the torus $T=S_1 \times S_1$. Choose two points $x,y \in T$ and define a quotient topology by identifying $x$ and $y$. Let $X$ denote the quotient space. Prove that: a) Compute the fundamental ...
22
votes
3answers
1k views

is the group of rational numbers the fundamental group of some space?

Which path connected space has fundamental group isomorphic to the group of rationals? More generally, is every group the fundamental group of a space?
7
votes
3answers
815 views

How to compute homotopy classes of maps on the 2-torus?

Let $\mathbb T^2$ be the 2-Torus and let $X$ be a topological space. Is there any way of computing $[\mathbb T^2,X]$, the set of homotopy class of continuous maps $\mathbb T^2\to X$ if I know, for ...
8
votes
1answer
250 views

Sanity check: Is every T-principal bundle over T trivial?

Is the following reasoning correct? The classifying space of the 1-torus $\mathbb T$ is $\mathbb{CP^\infty}$. Hence isomorphism classes of $\mathbb T$-principal bundles over $\mathbb T$ are in ...
3
votes
1answer
144 views

every map can be replaced by a weakly equivalent fibration

What is the meaning of the statement "every map can be replaced by a weakly equivalent fibration"?
20
votes
1answer
892 views

The Fundamental group of every subset of $\mathbb{R^2}$ is torsion free?

It seems that the fundamental group of any subset of $\mathbb{R^2}$ will not have an element of finite order. Though the 3-dimensional version is an open problem I couldn't immediately see why it is ...
-1
votes
3answers
485 views

rational homology [closed]

what's the isomorphism between $H_*(X;\mathbb Q)$ and $ H_*(X;\mathbb Z)\otimes \mathbb Q$
6
votes
1answer
975 views

universal cover of the sphere

what is the map $S^n \longrightarrow S^n$ that defines $S^n$ as the universal cover of $S^n$ ?
0
votes
1answer
73 views

r-skeleton of a geometric complex

I want to verify that the r-skeleton of a geometric complex is a geometric complex.This question was arised related with the topic geometric complexes and polyhedra.Can you help me? Besides this ...
3
votes
1answer
312 views

Constant maps from simplicial complexes to the n-sphere $S^n$

I've come unstuck and the following and was hoping someone could provide advice: X a space, triangulable as a simplicial complex with no n-simplices. Prove any cts map: $X \to S^n$ is homotopic to a ...
11
votes
1answer
291 views

What functor does $K(G, 1)$ represent for nonabelian $G$?

For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ ...
9
votes
3answers
946 views

Cohomology easier to compute (algebraic examples)

There is a previous post about motivating cohomology and it contains much of differential geometry examples, something I have just started and still have to figure out. It is said that one uses ...
2
votes
1answer
98 views

Operation on spaces and impact on algebraic invariants

I'm trying to get a feel for why different operations on spaces are useful. I realize this question is very long if someone wants to give a response to all the cases. With ''operations on spaces'' I ...
6
votes
0answers
216 views

Ring structure in the Serre spectral sequence

I've tried to understand what's going on in Example 1.5 on page 27-28 in Hatcher's notes on spectral sequences. There is one part in the reasoning that I can't understand here. He writes down a table ...
17
votes
1answer
552 views

Mapping class group vs outer automorphism group of the fundamental group for nonorientable surfaces

The Dehn--Nielsen--Baer theorem states that for a closed, connected and orientable surface M the extended mapping class group of M is isomorphic to the outer automorphism group of the fundamental ...
15
votes
1answer
199 views

Closure by Projective Limits of the category of Coverings of a Topological Space

Let $X$ be a connected topological space, and $C_{finite}$ the category of its finite coverings. Then I claim that the category $C$ of coverings of $X$ can be obtained by $C_{finite}$ taking ...
1
vote
1answer
482 views

Free groups and commutators

Good evening I was trying to prove that the commutator [F2,F2] of the free group F2 is not finitely generated by using covering spaces (i have to admit that this is the idea of a friend) it seems ...
3
votes
1answer
273 views

Octonionic Hopf fibration and $\mathbb HP^3$

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations ...
5
votes
1answer
280 views

Cartan 3-form on a Lie group G

Does anyone have a reference to learn more about the Cartan $3$-form on a group manifold $G$? I have read that the WZW Lagrangian is nothing more than the integral of the pullback of the Cartan ...
4
votes
1answer
114 views

On the functor H

Question: What can you deduce about $f$ by examining $H_{\ast}f?$ Detailed version of the question: Let $H$ be a homology theory which satisfy the Eilenberg-Steenrod-Milnor axioms (see, for ...
14
votes
3answers
3k views

What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...
3
votes
2answers
396 views

Classification of general fibre bundles

For principal $G$-bundles with $G$ a Lie group there exists a principal $G$-bundle $EG \to BG$ such that we have a bijection $$ [X,BG] \leftrightarrow \text{(principal $G$-bundles over X)} $$ $$ f ...
2
votes
2answers
571 views

Fundamental Group of a finite set with discrete topology

let S be a finite set with say n elements. Give it the discrete topology, Now what can we say about its fundamental group? Atleast can we determine the fundamental group of a set with two elements? ...
2
votes
1answer
68 views

Characterization of “Smallness” of the category of coverings over a topological space

Fixed a connected topological space $X$ it's an exercise to show that, if $X$ admits a universal covering $Y \rightarrow X$, then the category $C$ of finite covering spaces of $X$ is small. I'm ...
0
votes
1answer
142 views

What is the second stiefel whitney class of SO(n)?

$\omega_2(SO(n))=?$, that is, What is the second stiefel whitney class of SO(n)?
4
votes
0answers
750 views

How to calculate all the subgroups of the fundamental group of the Klein bottle?

A problem asks me to find all the covering spaces of a Klein bottle. This needs to calculate all the subgroups of the fundamental group of the Klein bottle. But I don't have any idea how to do it. I ...
1
vote
1answer
815 views

Why is a finite CW complex compact?

Hatcher explains on page 5 how a CW complex can be constructed inductively by attaching $n$-cells i.e. open $n$-dimensional disks. On page 520 in the appendix he writes "A finite CW complex, ... , is ...
15
votes
1answer
4k views

Why is a covering space of a torus $T$ homeomorphic either to $\mathbb{R}^2$, $S^1\times\mathbb{R}$ or $T$?

I know a sketch of the proof. M. A. Armstrong 's Basic Topology says that Suppose $X$ has a universal covering space, and denote it by $\tilde{X}$. Then the covering transformations form a group ...
8
votes
2answers
492 views

Examples of Computations in Algebraic Topology

I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing ...
2
votes
1answer
227 views

CW-Complexes Clarification: Finite Verses Infinite Cells in each Dimension

Question: Consider the following construction: take any uncountable set of elements and form a chain ($x < y < z <\dots$ ) of the elements (in particular, I think one can well-order the reals ...