Questions tagged [at.algebraic-topology]

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

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32
votes
15answers
9k views

“Homotopy-first” courses in algebraic topology

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
12
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3answers
2k views

Contractible manifold with boundary - is it a disc?

I'm sure this is standard but I don't know where to look. Let $M$ be a contractible compact smooth $n$-manifold with boundary. Does it have to be homeomorphic to $D^n$? What about diffeomorphic? [...
2
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2answers
412 views

homotopy type of complement of subspace arrangement

I am studying the homotopy type of a space,and i hope it would be a $K(\pi,1)$ space. now i have find its covering,once we can say the covering is $K(\pi,1)$,so is the space itself.and the covering is ...
13
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3answers
777 views

Homotopy type of set of self homotopy-equivalences of a surface

Let $\Sigma$ be an oriented topological surface. For simplicity, assume that the genus of $\Sigma$ is at least $2$. There are a number of classical results on the homotopy types of various groups of ...
7
votes
3answers
936 views

Cohomology rings of $ GL_n(C)$, $SL_n(C)$

Can anyone provide me with the reference for the following fact (idea of the proof will be appreciated too): Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what ...
2
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3answers
2k views

Homology with Coefficients

We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in ...
5
votes
3answers
386 views

PD3 groups and PD4 complexes

I am interested at the moment in what groups can occur as the fundamental group of a 4-manifold (or more generally, a 4-dimensional CW complex) with prescribed conditions on the intersection form. I ...
7
votes
1answer
360 views

How does this geometric description of the structure of PSL(2, Z) actually work?

There is a beautiful way to see that the congruence subgroup $\Gamma(2)$ is free on two generators: the action of $\Gamma(2)$ on $\mathbb{H}$ is free and properly discontinuous, and there is a modular ...
11
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2answers
682 views

Uniqueness of Chern/Stiefel-Whitney Classes

This question is closely related to this previous question. Chern and Stiefel-Whitney classes can be defined on bundles over arbitrary base spaces. (In Hatcher's Vector Bundles notes, he uses the ...
4
votes
4answers
2k views

circle action on sphere

surely $S^1$ can act on $S^n$ as a rotation.I want to know if there is some other way that a circle can act on sphere.
5
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0answers
475 views

Is the face poset a Heyting algebra?

Is the face poset of a simplicial or nice enough cellular complex a Heyting algebra in some natural way? Edited to add: For the benefit of illustration, here's a few face posets: the boundary of a ...
10
votes
2answers
1k views

Sheaves over simplicial sets

Is there a good way to define a sheaf over a simplicial set - i.e. as a functor from the diagram of the simplicial set to wherever the sheaf takes its values - in a way that while defined on simplex ...
23
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6answers
3k views

How should I visualise RP^n?

So I did some algebraic topology at university, including homotopy theory and basic simplicial homology, as well as some differential geometry; and now I'm coming back to the subject for fun via ...
11
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1answer
1k views

K-theory as a generalized cohomology theory

Which of the statements is wrong: a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point reduced complex $K$-theory $\tilde K$ and reduced real $K$...
46
votes
9answers
5k views

Fundamental groups of noncompact surfaces

I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
9
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1answer
1k views

Convergence of spectral sequences of cohomological type

Following the first chapter of Hatcher's great book "Spectral Sequences in Algebraic Topology", I got into problems with spectral sequences of cohomological type. Fix a ring $R$ once and for all. ...
-1
votes
1answer
792 views

Covering maps on Euclidean spaces and spheres [closed]

Hello. I have two questions. Does there exist an exactly 2-fold covering map $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ ? Does there exist an exactly 2-fold covering map $g:S^{n}\rightarrow S^{n}$ ? ...
8
votes
3answers
1k views

motivation of surgery

an $n$-surgery on m dim manifold M is to cut out $S^n\times D^{m-n}$and replace it by $D^{n+1}\times S^{m-n-1}$. I want to know how this is invented? I do know that the effect of passing a critical ...
7
votes
2answers
352 views

Relation between $KO$ and $K$

What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
30
votes
5answers
2k views

Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?

Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)? If not true in general, is there any condition ...
2
votes
1answer
801 views

Fixed points of continuous involutions of the plane

Hello. I would like to know how to prove that every continuous involution $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ (that is, $F(F(x))=x$ $\forall x \in \mathbb{R}^{2}$ ) has a fixed point? Thank you very ...
17
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1answer
681 views

Can we reconstruct positive weight invariants in algebraic topology using algebraic geometry?

I can't really say that I understand what a weight is, but the qualitative distinction between weight zero and positive weight has come up a couple times in MathOverflow questions: The étale ...
7
votes
4answers
731 views

Correspondences in Topology

I had one or two little fights with correspondences in the context of algebraic geometry where an elementary correspondence $C:X\to Y$ of connected smooth $k$-Schemes seems to be defined as an ...
49
votes
12answers
20k views

Algebraic Topology Beyond the Basics:Any Texts Bridging The Gap?

Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts ...
0
votes
1answer
330 views

Analogs of left, right, inner, and Kan fibrations in CGWH

It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. However, I know next to ...
6
votes
1answer
1k views

Mumford Conjecture

The Mumford Conjecture (now a theorem) says basically what is the (tautological subring)* of the rational cohomology ring of the stable moduli space of curves. Meaning that we know the ring structure (...
6
votes
1answer
1k views

The simplicial Nerve

The nerve functor $N:Cat\to SSet$ from the category of small categories to simplicial sets can be obtained as follows: The left Kan extension of the functor $F$ which sends $[n]$ to the category $\...
25
votes
8answers
4k views

Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
37
votes
6answers
8k views

Why are local systems and representations of the fundamental group equivalent

My question: Let X be a sufficiently 'nice' topological space. Then there is an equivalence between representations of the fundamental group of X and local systems on X, i.e. sheaves on X locally ...
27
votes
5answers
2k views

Small simplicial complexes with torsion in their homology?

Fix a prime $p$. What is the smallest integer $n$ so that there is a simplicial complex on $n$ vertices with $p$-torsion in its homology? For example, when $p=2$, there is a complex with 6 vertices (...
26
votes
5answers
3k views

Two-to-one continuous mapping from R² to R²

Hello. I have a question. Does there exist a continuous mapping $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that for every $c\in F(\mathbb{R}^2)$ there are two and only two points $z_{1}$, $z_{2}$...
1
vote
0answers
1k views

Again about Bing's house with two rooms [duplicate]

Possible Duplicate: How to show that the “bing’s house with two rooms” is contractible? I don't know why my question is closed? here, I make my question clearly, when "hollowing ...
3
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1answer
3k views

How to show that the “bing's house with two rooms” is contractible? [closed]

I can't image this, Someone can give a clear illustration?
9
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2answers
1k views

Is there a description of sheaf cohomology in algebraic-topological terms?

Is there a description of of sheaf cohomology for the sheaf of sections of a continuous function in terms of common constructions in Algebraic Topology? In more detail: Any sheaf on a space X can be ...
13
votes
2answers
502 views

Classifying spaces for enriched categories

Is there a standard construction of a classifying space $BC$ for a category $C$ which is enriched which takes into account the enrichment? This is of course vague... The simplest example I can think ...
12
votes
7answers
1k views

Cohomology classes annihilated by pullbacks

A friend of mine is interested in examples of the following situation: an oriented smooth fiber bundle $\pi \colon M \to B$ with $M$ and $B$ compact and a non-zero class $a \in H^3(B; \mathbb{Q})$ ...
37
votes
5answers
3k views

Why do wedges of spheres often appear in combinatorics?

Robin Forman writes in "A User's Guide to Discrete Morse Theory": The reader should not get the impression that the homotopy type of a CW complex is determined by the number of cells of each ...
6
votes
1answer
779 views

Serre spectral sequence with spectra

A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than ...
31
votes
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'\...
18
votes
2answers
5k 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
932 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
2k 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
votes
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
votes
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
2k views

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
401 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 ...