# Questions tagged [at.algebraic-topology]

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

5,989
questions

**32**

votes

**15**answers

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### “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**

votes

**3**answers

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**

votes

**2**answers

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**

votes

**3**answers

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

**3**answers

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**

votes

**3**answers

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

**3**answers

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

**1**answer

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**

votes

**2**answers

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

**4**answers

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**

votes

**0**answers

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

**2**answers

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**

votes

**6**answers

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**

votes

**1**answer

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### 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

**9**answers

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### 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**

votes

**1**answer

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

**1**answer

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

**3**answers

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### 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

**2**answers

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

**5**answers

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

**1**answer

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**

votes

**1**answer

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

**4**answers

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

**12**answers

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### 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

**1**answer

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

**1**answer

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

**1**answer

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

**8**answers

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

**6**answers

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

**5**answers

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

**5**answers

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

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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

**2**answers

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

**7**answers

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### 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

**5**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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### 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

**1**answer

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### 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

**5**answers

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

**4**answers

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

**3**answers

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

**5**answers

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

**1**answer

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 ...