# Questions tagged [at.algebraic-topology]

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

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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 ...
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### 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? [...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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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 $K$...
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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 ...
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### 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. ...
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### 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 $g:S^{n}\rightarrow S^{n}$ ? ...
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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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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 ...
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### 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 ...
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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 ...
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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 ...
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### 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, ...
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### 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 ...
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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 ...
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 ...