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

learn more… | top users | synonyms (1)

3
votes
1answer
349 views

Embedding torus in Euclidean space

For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?
12
votes
2answers
816 views

Why is Top a model category?

Recall that a model category is a complete and cocomplete category with classes of morphisms called cofibrations, fibrations, and weak equivalences. These are closed under composition and satisfy ...
2
votes
2answers
280 views

References for sheaf homology

Sheaf cohomology is a well-studied topic with a lot of references available. For example Hartshorne's book. But for a certain paper I am reading now, I have to understand sheaf homology. Could ...
16
votes
3answers
2k views

Meaning of relative homology

It is a bit easier to understand the homology $H_1(X, \mathbb Z)$ for various compact surfaces in analogy with handles and so on. There seems to be a nice intuitive picture with handles, holes, etc to ...
14
votes
1answer
794 views

Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras ...
6
votes
1answer
345 views

Is a map a homotopy equivalence if its suspension is so?

Let $X$, $Y$ be pointed CW complexes, $Y$ connected and $f:X\to Y$ a mapping. Does the assertion '$\Sigma f:\Sigma X\to\Sigma Y$ is a homotopy equivalence' imply that $f$ is a homotopy equivalence? ...
9
votes
1answer
780 views

Is the Serre spectral sequence a special case of the Leray spectral sequence?

Let $F \to E \to B$ be a fibration with $B$ simply connected (more generally, such that $\pi_1(B)$ acts trivially on the homology of $F$). Then there is a Serre spectral sequence $H_p(B, H_q(F)) \to ...
3
votes
1answer
193 views

Connecting Homomorphism in LES of fibration

Let $p:E\rightarrow B$ be a Serre fibration of path connected spaces with fiber $F$. Are the connecting homomorphisms $\partial:\pi_{n+1}(B)\rightarrow \pi_{n}(F)$ in the long exact sequence of $p$ ...
10
votes
4answers
635 views

Why is the cohomology of a $K(G,1)$ group cohomology?

Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular ...
6
votes
1answer
240 views

Why steenrod commute with transgression

I'm reading Hatcher's notes on spectral sequences and he mentions that steenrod squares commute with the coboundary operator for pairs (X,A) which would then explain why these operations commute with ...
4
votes
1answer
314 views

Quotient spaces and equivariant cohomology

Consider a $G$-equivariant map $\pi:X\to Y$ for $G$ an affine algebraic group, such that $\pi$ is a good categorical quotient. Is there any relationship between $H^*_G(X)$ and $H^*(Y)$? Is there if ...
5
votes
1answer
374 views

For what manifold is boundary given odd-dimensional projective space?

Take projective real space $\mathbb P_n (\mathbb R)$ of ODD dimension. It is easy to proof that all his Stiefel-Whitney numbers are zero . So according Thom theorem there must exists manifold $M$ with ...
2
votes
1answer
119 views

projective limit of simple groups

Let $G = \hbox{proj.lim.}_{\alpha} \{ G_{\alpha} , \varphi^{\alpha}_{\beta} \}$ be a projective limit of simple groups (i.e., each $\varphi^{\alpha}_{\beta}\colon G_{\alpha}\to G_{\beta}$ is a ...
1
vote
0answers
306 views

Computing degree of map

Suppose two manifolds $X$ and $Y$, both orientable of dimension $n$, and a map $f:X\to Y$. Is there a relationship between the degree of $f$ calculated with respect to homology (the induced map on ...
1
vote
1answer
125 views

Why is $H^n(I \times Y, R) \to H^n( \partial I \times Y, R)$ a split injection?

In Hatcher's Algebraic Topology, section 3.2, during the computation of the cohomology ring of a $n$-torus, the following assertion is made. Let $Y$ be a space and $R$ a commutative ring. Then the ...
9
votes
1answer
556 views

Why is stable equivalence necessary in topological K-theory?

The topological $K$-theory of a complex compact manifold $X$ is the commutative monoid $K(X)$ of isomorphism classes of complex vector bundles. Two classes $[E]$ and $[F]$ are equivalent in $K$-theory ...
0
votes
2answers
248 views

Degrees of certain maps of spheres

Suppose $S^n$ is the n-sphere with basepoint $x$. If based map $f:S^n\rightarrow S^n$ is such that the pull-back $f^{-1}(S^{n}-x)$ is connected do we necessarily have that $deg(f)$ is either -1,0, or ...
18
votes
8answers
1k views

Reference for spectral sequences

What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological ...
9
votes
1answer
405 views

Bott periodicity and algebraic geometry

It is a theorem that every locally free coherent sheaf on $\mathbb{P}^1$ over an algebraically closed field is isomorphic to a unique sum of sheaves $\mathcal{O}(n)$ for various integers $n$. In ...
5
votes
3answers
5k views

Homology of the Klein Bottle

I know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Also ...
5
votes
3answers
871 views

Topology of a cone of RP^2

I had already posted this on mathoverflow and was advised to post the same here. So here it goes: X={x,y,z|$x^2+y^2+z^2≤1$ and $ z≥0$} i.e. X is the top half of a 3-Disk. Z=X/E, where E is the ...
7
votes
2answers
852 views

Subgroups - Klein bottle

Let $G$ be the fundamental group of the Klein bottle, $G = \langle x,y \ ; \ yxy^{-1}=x^{-1} \rangle = {\mathbb Z} \rtimes {\mathbb Z} \ .$ What are the nilpotent subgroups of $G$? I was only ...
4
votes
2answers
524 views

Hairy Ball theorem and its applications

While searching a question about fibre bundles, which was asked here, i got directed to Vector bundles. I noticed this word "Hairy Ball" which sounded eccentric and made a search at Wikipedia. How ...
2
votes
1answer
679 views

Homology with local coefficients

Is there any relation between the homology of a space with local coefficients (in $\mathbb Q$ vector space) and the homology with coefficients in $\mathbb Q$? Thanks!
11
votes
1answer
294 views

Are homotopic maps over a cofibration homotopic relative to the cofibration?

Let $X$ be a Hausdorff space and $A$ a closed subspace. Suppose the inclusion $A \hookrightarrow X$ is a cofibration. Let $f, g: X \to Y$ be maps that agree on $A$ and which are homotopic. Are they ...
2
votes
3answers
230 views

How equivariant theory (like equivariant cohomology) arise

I understand in mathematics there are many "quotienting " proceduce, is this the only reason that we consider equivariant theory for different "unequivariant" theory? Are there any more applications ...
8
votes
1answer
293 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
14
votes
3answers
1k views

Integral classes in de Rham cohomology

If $M$ is a differentiable manifold, De Rham's theorem gives for each positive integer $k$ an isomorphism $Rh^k : H^k_{DR}(M,\mathbb R) \to H^k_{singular}(M,\mathbb R)$. On the other hand, we have a ...
13
votes
2answers
392 views

Finite generation in amalgamated free products

Let $G = A *_C B$ be an amalgamated free product of groups. My question is: suppose $C$ and $G$ are finitely generated, can we prove that so is $A$? I've been trying to prove it by contradiction. ...
13
votes
3answers
509 views

Connections between K-Theory and PDEs?

I've recently spent some time learning (the very basics of) K-theory for $C^*$-algebras and topological K-theory. Actually, my main fields of interest are PDEs and related topics, in particular ...
9
votes
2answers
1k views

Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group ...
3
votes
2answers
322 views

Does noncompact manifold or orbifold have the homotopy type,of CW complex?

I forget for a while, we don't need the compactness condition here right?
6
votes
1answer
262 views

Is there any relation about rational homology of X and X/G

If we know the rational homology of X is 0, can we get some information about the rational homology of X/G, where G is a finite group? Thank you very much for the answers!
3
votes
1answer
362 views

Why does this particular Serre spectral sequence collapse at the $E_{k+2}$-page?

The problem: Let $X$ be a product of equidimensional spheres of arbitrary dimension, say $k$, and $G$ a finite group acting freely on $X$. Assume that the induced $G$-action on the $Z_2$-cohomology ...
5
votes
1answer
232 views

Does contractibility imply contractibility with a basepoint?

Let $X$ be a contractible space. If $x_0 \in X$, it is not necessarily true that the pointed space $(X,x_0)$ is contractible (i.e., it is possible that any contracting homotopy will move $x_0$). An ...
4
votes
1answer
335 views

Euler class and Vandermonde polynomial

I found the following in the wikipedia page for Euler class. «If the rank $r$ is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$. Under the splitting ...
16
votes
3answers
2k views

Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?

I know, fundamental group of $GL(n,\mathbb{C})$ is isomorphic to $\mathbb{Z}$. It's written in Wikipedia. Actually, I've succeed in proving this, but my proof is two pages long and very technical. I ...
2
votes
2answers
288 views

Basic questions about the algebra of surfaces

When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double ...
372
votes
6answers
67k views

Why can you turn clothing right-side-out?

My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the ...
5
votes
4answers
4k views

Fundamental group of the double torus

In May's "A Concise Course in Algebraic Topology" I am supposed to calculate the fundamental group of the double torus. Can this be done using van Kampen's theorem and the fact that for (based) spaces ...
13
votes
12answers
2k views

Applications of algebraic topology

What are some nice applications of algebraic topology that can be presented to beginning students? To give examples of what I have in mind: Brouwer's fixed point theorem, Borsuk-Ulam theorem, Hairy ...
17
votes
1answer
2k views

Presentation of the fundamental group of a manifold minus some points

I recently noticed a few things in some recent questions on MO: 1) the fundamental group of $S^2$ minus, say, 4 points, is $\langle a,b,c,d\ |\ abcd=1\rangle$. 2) The fundamental group of a torus ...
11
votes
3answers
993 views

Toy sheaf cohomology computation

I asked this question a while back on MO : http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology One thing that really helped in learning the Serre SS ...
9
votes
5answers
548 views

Visualising $\mathbb CP^2$: a problem of attaching cells with a dimension gap >1

For the uninitiated Morse theory, as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, ...
6
votes
1answer
334 views

Varying definitions of cohomology

So I know that given a chain complex we can define the $d$-th cohomology by taking $\ker{d}/\mathrm{im}_{d+1}$. But I don't know how this corresponds to the idea of holes in topological spaces (maybe ...