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Questions tagged [at.algebraic-topology]

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

13
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5answers
989 views

What kind of geometric operations “scale up” cohomology?

There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant. Does anyone know of an operation on the level of ...
24
votes
4answers
3k views

(∞, 1)-categorical description of equivariant homotopy theory

I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
18
votes
5answers
1k views

Simplicial Model of Hopf Map?

The Hopf fibration is a famous map S3 --> S2 with fiber S1, which is the generator in pi_3(S2). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces ...
15
votes
8answers
2k views

How to get product on cohomology using the K(G, n)?

This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map ...
15
votes
3answers
3k views

Cohomology and Eilenberg-MacLane spaces

This question is related to this question from Dinakar, which I found interesting but don't yet have the background to understand at that level. Unless I'm mistaken, the rough statement is that $H^n(...
6
votes
2answers
783 views

One Point Compactification

Suppose X is a path-connected, locally compact, Hausdorff space and Y is its one-point compactification. Let G be the fundamental group of X and H be the fundamental group of Y. Is it true that the ...
2
votes
2answers
306 views

Classifying space of a crossed complex

Brown defines the classifying space of a crossed complex in the following way. Given a filtration X* of a space X, define the fundamental crossed complex by: C_0 = X_0, C_1=\pi(X_1,X_0) (the ...
3
votes
5answers
379 views

Classifying maps into homogeneous spaces up to homotopy

I'm still just a beginner in algebraic topology, but there's a specific problem I'd like to understand, which is how to classify maps from one space into another up to homotopy -- for instance, I've ...
30
votes
5answers
3k views

Some intuition behind the five lemma?

Slightly simplified, the five lemma states that if we have a commutative diagram (in, say, an abelian category) $$\require{AMScd} \begin{CD} A_1 @>>> A_2 @>>> A_3 @>>> A_4 @...
17
votes
7answers
3k views

Whitehead for maps

I made the following claim over at the Secret Blogging Seminar, and now I'm not sure it's true: Let $f: X \to Y$ and $g: X \to Y$ be two maps between finite CW complexes. If f and g induce the same ...
4
votes
2answers
1k views

(how) are vector bundles and homotopy groups related?

Hello, homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the ...
44
votes
12answers
17k views

Homological Algebra texts

I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe). As usual, ...
3
votes
7answers
3k views

de Rham Cohomology of surfaces

Does anyone know a good book where I can find the computation of the de Rham Cohomology of surfaces in R^3 and other classical manifolds (higher dimensional spheres and projective spaces for example) ?...
4
votes
1answer
176 views

Adapting families of diffeomorphisms to an open cover

Has anyone seen the following result in the literature? I've asked a few experts but so far I've come up with nothing. Given a manifold M and an open cover {U_i} ...
14
votes
5answers
3k views

How to determine the homotopy groups of the suspension of a space?

Let $SX$ be the suspension of CW complex. What are some results available to determine the homotopy groups of $SX$?
18
votes
5answers
2k views

Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
41
votes
11answers
4k views

What is the Cayley projective plane?

One can build a projective plane from R^n, C^n and H^n and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as OP^2, the Cayley projective plane....
6
votes
2answers
452 views

Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?

Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...
13
votes
2answers
803 views

Are generalized cohomology theories a homotopy category of some category of invariants?

I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors ...
2
votes
3answers
1k views

free homotopy groups — when do they exist?

Let (X,x) be a pointed space. There is an action of π1(X,x) on πn(X,x) -- determined by considering πn(X,x)=πn-1(ΩxX,x), where ΩxX denotes the space of loops in X based at x, ...
6
votes
1answer
266 views

Can you construct a mapping space from local data? (looking for reference)

I'd to know if/where there is a reference for the following construction. Let C_*(maps(M, T)) denote the singular chains on the space of continuous maps from an n-...
24
votes
3answers
2k views

Why is homology not (co)representable?

This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?
7
votes
3answers
844 views

Representablity of Cohomology Ring

I know that the individual cohomology groups are representable in the homotopy category of spaces by the Eilenberg-MacLane spaces. Is it also true that the entire cohomology ring is representable? If ...
2
votes
2answers
1k views

proving that an inclusion map from a subcomplex is a homotopy equivalence

This is a pretty basic question but I have been stuck on it for a while. Given an abstract simplicial complex X and a subcomplex A, why does * suffice to show that the map |A|->|X| induced by ...
6
votes
1answer
494 views

Pontryagin product from an operad

For a topological group G, we have a Pontryagin product in homology by multiplying representative cycles. This gives the homology the structure of an associative graded algebra. Am I correct in ...
3
votes
3answers
1k views

singular cohomology of SO(4)

I'm trying to compute the singular cohomology of SO(4), just as practice for using spectral sequences. I got H0=Z, H1=0, H2=Z/2Z, H3=Z⊕Z, H4=0, H5=Z/2Z, and H6=Z. Are these correct? I'm not ...
14
votes
8answers
2k views

Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
13
votes
1answer
670 views

Commutativity in K-theory and cohomology

The Chern classes give a map $f : BU \to \prod_n K(\mathbb{Z},2n)$, which is a rational equivalence. However, it is not an equivalence over $\mathbb{Z}$ because the cohomology of $BU$ is just a ...
15
votes
4answers
2k views

How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
35
votes
5answers
4k views

Analogue to covering space for higher homotopy groups?

The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched ...
16
votes
5answers
4k views

Describing the universal covering map for the twice punctured complex plane

As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map. In a sense, this shows that the logarithm has ...
17
votes
2answers
3k views

Euler characteristic of a manifold and self-intersection

This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ...
33
votes
7answers
3k views

Simplicial objects

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
6
votes
2answers
857 views

Differentials in the Lyndon-Hochschild spectral sequence

The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration. Does anyone know of a good description (...
60
votes
9answers
9k views

understanding Steenrod squares

There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
45
votes
5answers
4k views

Does homology have a coproduct?

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
80
votes
10answers
17k views

Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
9
votes
2answers
962 views

Elliptic curve over spectra?

Filling the gaps in my knowledge to understand the tmf question. So, what is the analogue of elliptic curve over the category of spectra?
32
votes
6answers
11k views

How do you show that $S^{\infty}$ is contractible?

Here I mean the version with all but finitely many components zero.
10
votes
5answers
3k views

Ribbon graph decomposition of the moduli space of curves

What is a ribbon graph? What is the ribbon graph decomposition of the moduli space of curves? What are some good references for this material?
33
votes
5answers
4k views

How can you tell if a space is homotopy equivalent to a manifold?

Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
0
votes
0answers
1k views

Ignore this question [closed]

This question is a hacky way to create some tags for you to use. Move along.