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)

4
votes
1answer
235 views

What's the map $BU \times \mathbb{Z} \to \prod K(\mathbb{Z},n)$ representing the total Chern class?

Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total ...
7
votes
1answer
404 views

Hopf invariant formula

I'm having some trouble proving the following statement: let $g:S^{2n-1} \to S^{2n-1}, f: S^{2n-1} \to S^n$. Then $H(f\circ g) = \deg g H(f)$ where $H(f)$ is the Hopf invariant. The definition I am ...
1
vote
1answer
168 views

Construct CW complex

How can I construct a CW complex A with $H_0(A) = \mathbb{Z}$, $H_2(A) = \mathbb{Z}/4\mathbb{Z}$, $H_4(A) = \mathbb{Z}\oplus\mathbb{Z}$ and all other homology groups trivial? Any idea? Thanks!
13
votes
2answers
604 views

Question about the first proof in Hatcher's Algebraic Topology

I have a question about Hatcher's proof that the fundamental group of a circle is Z. Specifically, halfway through, ( http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf , page 30), he proves an ...
11
votes
5answers
1k views

Poincare Duality Reference

In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's ...
0
votes
1answer
162 views

Homotopy of two paths whose composition is nullhomotopic

Let $f, g$ be two paths in a space such that their concatenation $f * g$ is nullhomotopic. Prove that $f$ is homotopic to $g$ rel $\{0,1\}$.
5
votes
1answer
204 views

Ring structure of $H^2(S^2 \vee S^4)$

We know that $H^p(S^2 \vee S^4) = H^p(S^2)\oplus H^p(S^4)$ for $p\neq 0$. I want to show that this space has different ring structure than $CP^2$. So, given a generator in $H^2(S^2 \vee S^4)$ I want ...
18
votes
4answers
1k views

Motivating Cohomology

Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology? Why ...
13
votes
1answer
719 views

Fundamental group of quotient spaces of $SO(3)$

I am trying to figure out the fundamental group (actually simply connected or not will suffice) of the following quotient space of $SO(3)$: Let $X = SO(3)/E$, where $E$ is the equivalence relation ...
13
votes
4answers
1k views

Applications for Homology

The Question: Are there any ways that "applied" mathematicians can use Homology theory? Have you seen any good applications of it to the "real world" either directly or indirectly? Why do I care? ...
3
votes
4answers
327 views

Chromatic Filtration of Burnside Ring

I just attended a seminar on the chromatic filtration of the Burnside ring. I understood it relatively well, but at no point did anyone give an explicit definition of what a chromatic filtration ...
13
votes
3answers
523 views

How to prove a manifold is simply connected?… using geometry

I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question. ...
9
votes
1answer
2k views

What local system really is

I know a local system is a locally constant constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, ...
10
votes
2answers
761 views

Homology and Euler characteristics of the classical Lie groups

I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques ...
4
votes
0answers
395 views

Is every CW complex homotopic to a Delta-Complex?

Both answers to this question seem equally reasonable to me. If the answer is positive, I have no idea what the construction of such a space would look like.... If the answer is negative, I assume ...
2
votes
1answer
154 views

Blow-up. Graph of canonical map: is a complex manifold no closed or a topological space no closed

I am reading by myself this book http://tinyurl.com/37z4bbt. But to be honest, I have several problems to fully understand some part of the text. Maybe because I have not yet solid knowledge or I ...
3
votes
1answer
484 views

Relative Cohomology Isomorphic to Cohomology of Quotient

Given a topological space (with nice enough conditions, maybe Hausdorff, compactly generated, or CW complex, I'm not sure) $X$ and a subspace $A\subset X$, is it true that $H^n(X,A)\cong ...
3
votes
1answer
340 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 $\mathbb R^{n+1}$?
11
votes
2answers
737 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
255 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
758 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
318 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
705 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
183 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
583 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
225 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
290 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
349 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
117 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
285 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
117 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
513 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
242 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 ...
17
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 ...
8
votes
1answer
394 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
4k 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
795 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
814 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
489 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
634 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
285 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
227 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
288 views

How does hocolim relate to Hom?

In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like ...
13
votes
3answers
918 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
375 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
485 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 ...
8
votes
2answers
938 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
294 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
256 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!