Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
1,148
questions
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Early stabilization in the homotopy groups of spheres
Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed ...
- 13.2k
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votes
2
answers
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Converse to Stokes' Theorem
Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : \Lambda^...
- 61.5k
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votes
7
answers
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Are there two non-homotopy equivalent spaces with equal homotopy groups?
Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth ...
- 2,459
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votes
3
answers
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What are the higher homotopy groups of a K3 suface?
All K3 surfaces have the same homotopy type. What are their higher homotopy groups?
I know that $\pi_1$ is trivial, and $\pi_2$ is $\mathbb{Z}^{22}$.
Even if the answer isn't known in all degrees, I'...
- 15.1k
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votes
1
answer
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For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?
My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
- 115k
29
votes
6
answers
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Concrete example of $\infty$-categories
I've seen many different notions of $\infty$-categories: actually I've seen the operadic-globular ones of Batanin and Leinster, and the opetopic, and eventually I'll see the simplicial ones too. ...
- 3,299
28
votes
5
answers
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Modern source for spectra (including ring spectra)
I am looking for a modern introduction to spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what ...
28
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4
answers
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Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
- 4,017
27
votes
3
answers
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Configuration space of little disks inside a big disk
The space of configurations of $k$ distinct points in the plane
$$F(\mathbb{R}^2,k)=\lbrace(z_1,\ldots , z_k)\mid z_i\in \mathbb{R}^2, i\neq j\implies z_i\neq z_j\rbrace$$
is a well-studied object ...
- 35k
26
votes
3
answers
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"Dirty" proof that Eilenberg-MacLane spaces represent cohomology?
The standard approach to proving that $H^n(X; G)$ is represented by $K(G, n)$ seems to be to prove that $\text{Hom}(X, K(G, n))$ defines a cohomology theory and then use Eilenberg-Steenrod uniqueness. ...
- 2,148
26
votes
3
answers
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Why is BG infinite dimensional for G finite ?
If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$
then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This
can be found, for example, there:
Non-...
- 2,170
26
votes
3
answers
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views
Third differential in Atiyah Hirzebruch spectral sequence
Does any one know why $d_3: H^* (X, K^0(point))\rightarrow H^{*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z} $ coefficient.
- 883
26
votes
1
answer
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connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,084
25
votes
2
answers
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generalisations of the Seifert-van Kampen Theorem?
I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.
My attention was ...
- 12.2k
25
votes
1
answer
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From the perspective of bordism categories, where does the ring structure on Thom spectra come from?
To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
- 115k
23
votes
2
answers
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complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
23
votes
3
answers
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Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 1,095
23
votes
2
answers
3k
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Calculating Mayer-Vietoris efficiently
This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
- 151k
23
votes
1
answer
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The Dedekind eta function in physics
This interesting little fellow (a nice introduction is the video "Mock Modular Forms are Everywhere" by Cheng and Felder) popped up in some operator algebra (Witt / Virasoro Lie algebra) I ...
22
votes
1
answer
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When is a quasi-isomorphism necessarily a homotopy equivalence?
Under what circumstances is a quasi-isomorphism between two complexes necessarily a homotopy equivalence? For instance, this is true for chain complexes over a field (which are all homotopy ...
- 10k
22
votes
6
answers
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cohomology of BG, G compact Lie group
It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group
$G$. However, I've still not found a proof of this. I believe that the proof is as follows:
--> $G$ compact ...
- 1,699
22
votes
6
answers
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Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
- 1,264
22
votes
1
answer
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Word maps on compact Lie groups
Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$.
Murray Gerstenhaber and ...
- 25.3k
22
votes
5
answers
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Homeomorphisms of $S^n\times S^1$
Is every homeomorphism $h$ of $S^n\times S^1$ with identity action in $\pi_1$ pseudo isotopic to a homeomorphism $g$ of $S^n\times S^1$ such that $g(S^n\times x)=S^n\times x$ for each $x\in S^1$? I ...
- 339
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votes
1
answer
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Word problem for fundamental group of submanifolds of the 4-sphere
Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$. For most constructions of these 4-...
- 43.1k
22
votes
1
answer
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Little disks operad and $Gal (\bar {Q}/Q)$
My question is simple:
How do the little disks operad and $Gal (\bar {Q}/Q)$ relate?
I realize that a huge amount of heavy-machinery can be brought into an answer to this, but I'm struggling with ...
- 2,714
21
votes
2
answers
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When is a topological space the homotopy colimit of an open covering?
Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...
- 15.3k
21
votes
5
answers
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Maps inducing zero on homotopy groups but are not null-homotopic
Today my fellow grad student asked me a question, given a map f from X to Y, assume $f_*(\pi_i(X))=0$ in Y, when is f null-homotopic?
I search the literature a little bit, D.W.Kahn
Link
And M....
- 1,150
21
votes
3
answers
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Why is Kan's $Ex^\infty$ functor useful?
I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful ...
- 61.5k
21
votes
2
answers
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Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type ...
- 489
21
votes
4
answers
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Multiplicative Structures on Moore Spectra
The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
- 2,277
21
votes
2
answers
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Topologically contractible algebraic varieties
From a post to The Jouanolou trick:
Are all topologically trivial (contractible) complex algebraic varieties necessarily affine? Are there examples of those not birationally equivalent to an affine ...
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20
votes
2
answers
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The first unstable homotopy group of $Sp(n)$
Thanks to the fibrations
\begin{align*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align*}
we know that
\begin{align*}
\pi_i(SO(...
- 18.8k
20
votes
3
answers
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views
Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$?
Any element of $H^1(M,\mathbb{Z}/2)$ is the $w_1(E)$ of a real line bundle $E$ over $M$.
I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are ...
- 6,063
20
votes
3
answers
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Non-stably trivial bundle with trivial characteristic classes
Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not ...
- 4,342
20
votes
1
answer
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The cell structure of Thom spectra
I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...
- 6,218
20
votes
4
answers
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Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...
- 373
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3
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Simultaneous "orthonormalization" in $\mathbb{C}^4$
Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix.
So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
- 42.1k
19
votes
3
answers
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Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?
Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact ...
- 9,236
19
votes
1
answer
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When does the free loop space fibration split?
This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be ...
- 2,270
19
votes
4
answers
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Details for the action of the braid group B_3 on modular forms
I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...
- 115k
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votes
5
answers
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How do you define the strict infinity groupoids in Homotopy Type Theory?
In the setting of Homotopy Type Theory, how would you construct $\mathrm{isStrict} : U \rightarrow U$ which is inhabited exactly when the first type is (equivalent to?) a strict $\infty$-groupoid?
...
- 27.8k
19
votes
4
answers
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What are the fibrant objects in the injective model structure?
If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
- 27.2k
18
votes
3
answers
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Are finite spaces a model for finite CW-complexes?
Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, ...
- 42.5k
18
votes
1
answer
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Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups
Let $X$ be a simply connected finite CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily ...
- 11.9k
18
votes
2
answers
2k
views
Homotopy types of schemes
Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...
- 15.3k
18
votes
3
answers
2k
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Can eta invariant be written in terms of topological data?
The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In the paper "Exotic ...
- 875
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votes
3
answers
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When does the tangent bundle of a manifold admit a flat connection?
Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$?
Edit: Formerly, I asked about a flat connection on the frame ...
- 8,392
18
votes
3
answers
3k
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When are (finite) simplicial complexes (smooth) manifolds?
Hi,
is there an algorithm that determines if a given simplicial complex is
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?
- 939
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votes
3
answers
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Is there Domain Invariance for Alexandrov spaces?
A colleague asked me this question recently. Every injective continuous map between manifolds of the same (finite) dimension is open - this is Brouwer's Domain Invariance Theorem. Is the same true for ...
- 32.1k