Questions tagged [at.algebraic-topology]

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

5,790 questions
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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 ...
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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 ...
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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(... 2answers 796 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 ... 2answers 311 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 ... 5answers 383 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 ... 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 @... 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 ... 2answers 2k 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 ... 12answers 18k 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, ... 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) ?... 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} ... 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$SX$? 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 ... 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.... 2answers 455 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 ... 2answers 812 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 ... 3answers 2k 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, ... 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-... 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? 3answers 861 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 ... 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 ... 1answer 504 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 ... 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 ... 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 ... 1answer 677 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 ... 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 ... 5answers 5k 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 ... 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 ... 3answers 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 ... 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 [... 2answers 863 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 (... 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 ... 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 ... 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 ... 2answers 978 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? 6answers 12k views How do you show that$S^{\infty}\$ is contractible?

Here I mean the version with all but finitely many components zero.
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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?