Questions tagged [at.algebraic-topology]

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

31
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3answers
4k views

What is so “spectral” about spectral sequences?

From recent mathematical conversations, I have heard that when Leray first defined spectral sequences, he never published an official explanation of his terminology, namely what is "spectral" about a ...
3
votes
1answer
1k views

Fiber bundle = principal bundle + fiber?

This question is heavily related to this question. Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
17
votes
2answers
4k views

Group completion theorem

Let $M$ be a topological monoid. How does the homology-formulation of the group completion theorem, namely (see McDuff, Segal: Homology Fibrations and the "Group-Completion" Theorem) If $\pi_0$ is ...
14
votes
1answer
920 views

Complex orientations on homotopy

I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It ...
8
votes
2answers
1k views

Chas-Sullivan string topology

I recently read the original paper by Chas-Sullivan on string topology, in which they introduce some operations on homology of free loopspace LM, where M is a compact oriented manifold, giving it the ...
8
votes
1answer
1k views

homotopy associative $H$-space and $coH$-space

Let $[X, Y]_0$ denote base point preserving homotopy classes of maps $X\rightarrow Y$. A multiplication on a pointed space $Y$ is a map $\phi: Y\times Y\rightarrow Y.$ From this map, we can define a ...
9
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1answer
2k views

De Rham homology

Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities. On the category of finite-dimensional vector bundles over M and linear differential operators between them there is a ...
17
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5answers
4k views

Stiefel-Whitney Classes over Integers?

An interesting thing happened the other day. I was computing the Stiefel-Whitney numbers for $\mathbb{C}P^2$ connect sum $\mathbb{C}P^2$ to show that it was a boundary of another manifold. Of course, ...
17
votes
4answers
1k views

Cohomology of a sheaf of functions locally constant along a foliation

Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology ...
12
votes
3answers
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Applications of homotopy groups of spheres

The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is ...
29
votes
5answers
3k views

Topologists loops versus algebraists loops

Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also ...
3
votes
1answer
399 views

Principal bundle for contractible group is weak homotopy equivalence for ind schemes

This is may be obvious, but I am not comfortable with ind-schemes. I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
4
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0answers
337 views

homotopy groups of cubical sets

A cubical set $Box^{op} \to Set$ is a model for a homotopy type, via Grothendieck and Cisinski (here $Box$ is the box category with objects the natural numbers and arrows generated by face and ...
11
votes
4answers
3k views

homology with compact supports

In one of the exercises in McDuff and Salamon, they mention homology with compact supports. I know how to define *co*homology with compact supports, but I can't picture the homology version. How do ...
6
votes
1answer
701 views

Spectra and localizations of the category of topological spaces

Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical ...
14
votes
3answers
1k views

$(\infty,1)$-categories and model categories

I read several times that $(\infty,1)$-categories (weak Kan complexes, special simplicial sets) are a generalization of the concept of model categories. What does this mean? Can one associate an $(\...
7
votes
2answers
1k views

Intersection form in twisted homology (homology with local coefficients)

The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
8
votes
2answers
678 views

Is the cohomology of a topological operad a cooperad?

For cohomology with coefficients in a field $F$ the map $H^\cdot(X;F) \otimes H^\cdot(Y;F) \to H^\cdot(X \times Y;F)$ of the Kunneth theorem is an isomorphism of algebras over $F$. I am correct in ...
2
votes
1answer
432 views

How to build the principal SU(2) bundles on surfaces?

Is there a way to classify (and build) the principal SU(2) bundles over a given topological surface up to homeomorphism? In the end, I would like to examine the associated bundle whose fiber is a ...
19
votes
5answers
1k views

Is any interesting question about a group G decidable from a presentation of G?

We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
7
votes
1answer
705 views

Whitehead Products without Base Points?

Let $(X, x_0)$ be a pointed space. Then we can define the homotopy groups $\pi_i(X, x_0)$ for $i \geq 1$. They are abelian groups for $i \geq 2$. It is well-known that the fundamental group $\pi_1(X, ...
2
votes
1answer
493 views

characterization of cofibrations in CW-complexes with G-action

Is there a condition for a $G$-equivariant map $X \to Y$ to be a cofibration of $G$-spaces? Here $X$ and $Y$ are CW complexes, the group $G$ is finite, and acts by cellular maps. I am using the model ...
5
votes
3answers
961 views

Classification of simply connected smooth projective varieties?

This question is related to this one. I am wondering whether there is any sort of classification of simply connected smooth projective varieties, or any work in related directions. The reason I am ...
7
votes
3answers
939 views

Group cohomology vs. topological cohomology in the case of non-trivial action

When A is an abelian group with trivial G-action (G being a discrete group) we get that Hn(G,A)≅Hn(BG,A). Is there a similar connection between group cohomology and topological cohomology if A ...
35
votes
7answers
5k views

What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
7
votes
1answer
638 views

Surgery and homology: a reference request

I need a reference (or a short proof) for the following statement: Suppose a closed manifold $N$ is the result of a surgery (along an embedded sphere) on a closed manifold $M$. Then the difference $\...
35
votes
3answers
2k views

What is the classifying space of “G-bundles with connections”

Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
1
vote
2answers
866 views

Simple question of topological cofibration

I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
1
vote
1answer
223 views

blowing up the graphs

I heard the phrase from many mathematician using in the colloquials. I heard algebraic geometer using it. I was never bother about it until one of my professor responded to one my question as follows: ...
25
votes
3answers
3k views

finite generated group realized as fundamental group of manifolds

This is discussed in the standard textbooks on algebraic topology. Pick a presentation of the group $G = \langle g_1,g_2,...,g_n|r_1,r_2,...r_m \rangle$ where $g_i$ are generators and $r_j$ are ...
7
votes
1answer
534 views

Whitehead products on manifolds

What are some good examples of simply connected manifolds with interesting Whitehead Lie algebras over R? Most of the manifolds that one thinks about if one is pretty naive are not so interesting--- ...
13
votes
2answers
537 views

How do you compute the space of lifts of an E-infinity map?

Let X, Y and B be $E_\infty$ spaces, and let $p: X \rightarrow Y$ and $f: B \rightarrow Y$ be $E_\infty$ maps. We can ask for the space of lifts of f across p, that is the space of $E_\infty$ maps $g:...
9
votes
3answers
2k views

How are fiber bundles, transition functions and principal bundles related?

Please read the edit below. Is my understanding of this correct? Fix a sufficiently nice and connected topological space $B$ and a topological group $G$. A principal bundle $E\to B$ with structure ...
6
votes
1answer
590 views

primitive of an exact differential form with special properties

We were working on a smoothing problem and ran across the apparently simple following question: X is a triangulated smooth manifold of dimension $n$, and $\alpha$ is an exact differential form of ...
6
votes
2answers
760 views

visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...
10
votes
4answers
1k views

triviality of fibre bundles

are there some general method in judging if a fible bundle is trivial? At least,for vector bundles,there is a well-developed theory,that is Charicteristic Classes.the triviality of vector bundles is ...
3
votes
1answer
327 views

When are two natural transformations of infinity-categories equivalent?

Suppose C and D are two ∞-categories (quasi-categories), $F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the ...
6
votes
1answer
610 views

Some questions on the intersection theory on a Hilbert scheme of points of a surface.

If $\Sigma$ is a smooth complex curve in a smooth projective surface $X$, then we can consider the homology class represented by $\Sigma^{[n]} \subset X^{[n]}$. $\ \ $ Where, $X^{[n]}, \Sigma^{[n]}$ ...
32
votes
9answers
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Computing fundamental groups and singular cohomology of projective varieties

Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations ...
2
votes
2answers
867 views

What can be said about the homotopy groups of a CW-complex in terms of its (co)homology?

One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H_1 is nontrivial. What other examples are there? (The CW-complexes I ...
5
votes
3answers
2k views

Topological description of Manifold with boundary

Is there any 'general' topological invariant to tell the difference between $M$ and $N$, where $M$ has homotopy type of a closed manifold and $N$ has homotopy type of a manifold with boundary. I ...
2
votes
2answers
544 views

Must a Strong deformation retractible 3-manifold be homeomorphic to $\mathbb{R}^3$?

Assume $M$ is an open 3-manifold which can be deformation retracted to a point. Is it necessarily homeomorphic to $\mathbb R^3$? (I know Whitehead had an example which is contractible and not ...
5
votes
1answer
250 views

how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?

So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
20
votes
1answer
2k views

Every Manifold Cobordant to a Simply Connected Manifold

I am wondering if it is true that every compact, connected, oriented manifold is cobordant to a simply connected manifold. I believe that some sort of surgery will do the trick. Roughly speaking, I ...
9
votes
2answers
927 views

Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?

Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of quivers)...
16
votes
3answers
1k views

How does $\pi_1(SO(3))$ relate exactly to the waiters trick?

I hope this is serious enough. It is a well-known fact that $\pi_1(SO(3)) = \mathbb{Z}/(2)$, so $SO(3)$ admits precisely one non trivial covering, which is 2-sheeted. Another well known fact is that ...
16
votes
4answers
2k views

Checking if two graphs have the same universal cover

It's possible I just haven't thought hard enough about this, but I've been working at it off and on for a day or two and getting nowhere. You can define a notion of "covering graph" in graph theory, ...
6
votes
2answers
526 views

An algebraic proof of Mumford's smoothness criterion for surfaces?

(Disclaimer: I'm a beginner in this area, so welcome corrections.) Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
17
votes
6answers
5k views

PDE on manifolds

I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE ...
16
votes
2answers
871 views

What manifold has $\mathbb{H}P^{odd}$ as a boundary?

This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting. ...