Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
2,269
questions with no upvoted or accepted answers
14
votes
0
answers
366
views
When is a map of topological spaces homotopy equivalent to an algebraic map?
My question is simple, but I don't expect there are any simple answers.
Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....
- 1,635
14
votes
0
answers
238
views
How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$
Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...
- 549
14
votes
0
answers
328
views
Are there Alexander-Whitney maps in geometric homology?
When $X$ is a smooth manifold, Lipyanskiy defines a chain complex whose homology is isomorphic to singular homology -
let's say $GC_*(X)$ - generated by maps $\sigma: M \to X$ from compact $k$-...
- 9,388
14
votes
0
answers
270
views
Homotopy type of spaces of functions with few critical points
Given a closed manifold $M$ and an integer $k\geq 0$, let $G_k(M)$ denote the space of smooth functions $f:M\to\mathbb R$ with at most $k$ critical points.
To what extend has the topology of the ...
- 18.3k
14
votes
0
answers
516
views
Reference for equivariant Atiyah-Jänich theorem
The equivariant Atiyah-Jänich theorem is an isomorphism
$$
[X,F]_G \cong K_G^0(X),
$$
where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a ...
- 4,304
14
votes
0
answers
410
views
Does the category of G-spectra know G?
I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
- 8,643
14
votes
0
answers
305
views
Uniqueness of connected cover of Morava K-theory
Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$
where $A$ is the Steenrod algebra and $Q_n$ is ...
- 2,013
14
votes
0
answers
786
views
What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?
Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...
- 10.1k
14
votes
0
answers
545
views
Reference for a proof of the fiberwise Stokes theorem
The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...
- 36.3k
14
votes
0
answers
634
views
Who stated and proved the "Hopf lemma" on bilinear maps?
If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.
Nondegenerate here means that ...
- 1,801
14
votes
0
answers
769
views
How to see the quaternionic hopf map generates the stable 3-stem?
I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example:
third-stable-...
- 27.2k
14
votes
0
answers
2k
views
Homotopy groups of the orthogonal group
I'm interested in knowing what $n$-dimensional vector bundles on the $n$-sphere look like, or equivalently in determining $\pi_{n-1}(SO(n))$; here's a case that I haven't been able to solve.
Let $n \...
- 25.3k
14
votes
1
answer
2k
views
Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
- 2,450
13
votes
0
answers
373
views
Does the existence of an almost complex structure solely depend on the topology of the manifold?
To be precise, let $M$ and $N$ be two 2n-dimensional smooth, closed manifolds that are homeomorphic. If $M$ admits an almost complex structure, can we deduce that $N$ also admits an almost complex ...
- 619
13
votes
0
answers
221
views
Examples of manifolds with first nontrivial SW-class in degree 16 or bigger
As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
- 11.9k
13
votes
0
answers
289
views
The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
- 10.1k
13
votes
0
answers
860
views
A step in Toda's computation of a Cotor
I am trying to understand a proof from Toda's paper Cohomology of classifying spaces. The step I am stuck on is at page 96. Here is the setup.
We work with cohomology with $\mathbb{F}_2$ coefficients. ...
- 31
13
votes
0
answers
331
views
Morava K-theory of loop spaces of spheres
Some time ago I cam across the paper "What we still don't know about loop spaces of spheres" by Ravenel:
https://people.math.rochester.edu/faculty/doug/mypapers/loop.pdf
which concerns ...
- 1,759
13
votes
0
answers
441
views
Structures between PL and smooth
Let $X$ be a topological manifold of dimension at least five. The Kirby-Siebenmann invariant $ks(X)\in H^4(X,\mathbb{Z}_2)$ is an obstruction to the existence of a PL structure on $X$. If it vanishes, ...
- 1,493
13
votes
0
answers
317
views
Exotic smooth structures on Fano manifolds
If two Fano projective manifolds are homeomorphic are they diffeomorphic?
There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user164740
13
votes
0
answers
323
views
One periodic cohomology theories?
Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. ...
- 5,201
13
votes
0
answers
395
views
Computations using "Stover's spectral sequence"
In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.
The second ...
- 198
13
votes
0
answers
440
views
Examples of non-proper model structure
I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
- 40.2k
13
votes
0
answers
317
views
How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?
Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
- 53.1k
13
votes
0
answers
290
views
A geometric interpretation of the odd-primary Kervaire elements
Let $\Omega^\mathrm{fr}_\ast \cong \pi_\ast S$ denote the graded ring of cobordism classes of framed manifolds, which, by the Pontryagin-Thom construction, is isomorphic (as a graded ring) to the ...
- 5,550
13
votes
0
answers
259
views
Alexander modules and weight filtrations
$\def\Ext{\mathrm{Ext}}$I have the following situation: I have a complex algebraic variety $X$. I want to compute the weight filtration on various abelian covers $Y \to X$. As a concrete example, take ...
- 151k
13
votes
0
answers
530
views
Cohomology of a blow-up of a real algebraic variety
Let $X$ be a complex algebraic variety, $Z \subset X$ a closed subvariety, $\mathrm{Bl}_Z X$ the blow-up and $E$ the exceptional divisor. There is an isomorphism of cohomology groups
$$ H^k(X(\mathbf ...
- 39.3k
13
votes
0
answers
283
views
Actions of $\mathbb Z/2\mathbb Z$ on algebraically closed fields and even-dimensional spheres and parallel between Galois theory and covering theory
It is well known that there is a parallel between Galois theory and covering theory. So I wonder whether there is a deep similarity between the following two facts:
Artin-Schreier theorem. The only ...
- 1,990
13
votes
0
answers
365
views
What is the cup-product structure like on a hyperbolic 5-manifold?
Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?
For example, are there hyperbolic 5-manifolds ...
- 4,849
13
votes
0
answers
584
views
Finite groups inside an infinite group with the same homology
Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:
$G$ and $H$ are finite groups and $K$ is an infinite group.
there exist two monomorphisms $G \rightarrow K \leftarrow H$...
- 1,974
13
votes
0
answers
241
views
Is every simply connected finite complex the classifying space of a finite monoid
On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
- 655
13
votes
0
answers
550
views
When does an $E_\infty$ algebra come from a commutative differential graded algebra?
Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...
- 15.3k
13
votes
0
answers
1k
views
Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$
Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and $...
- 131
13
votes
0
answers
452
views
Singular cohomology of $BG$ and Borel cohomology of $G$
Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner.
For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, $\...
- 9,388
13
votes
0
answers
406
views
Does the de Rham version of Cohen's theorem hold in the $\infty$-setting?
One of the first results that one needs to prove in the theory of chiral algebras is a de Rham version of Cohen's theorem on the homology of $C_n$ spaces. This is achieved in Beilinson-Drinfeld's book ...
- 3,192
13
votes
0
answers
406
views
Topological type of Brieskorn manifolds
Let us consider the complex hypersurface and suppose that $n\geq 3$:
$$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$
and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ ...
- 9,802
13
votes
0
answers
664
views
Singular chains generated by manifolds with corners --- does it really work?
Usually we define singular homology via the complex of singular chains:
$$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$
where the right hand side denotes the free abelian ...
- 18.3k
13
votes
0
answers
848
views
About maps inducing bijections on homotopy classes
Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
- 20.6k
12
votes
0
answers
219
views
Is there an analogue of Steenrod's problem for $p>2$?
An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be realisable if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The ...
- 317
12
votes
0
answers
448
views
Proofs of Serre's theorem on simply-connected finite CW complexes
A famous result due to Serre states that any simply-connected finite CW complex with non-trivial $\mathbb{Z}_2$ homology has infinitely many non-zero homotopy groups. (In fact, Serre proves more than ...
12
votes
0
answers
219
views
Spin 6-fold with signature $\pm 16$
Does there exist a spin (i.e. $\frac{c_{1}}{2} \in H^{2}(M,\mathbb{Z})$) smooth complex projective $6$-fold with signature $\pm 16$?
The motivation is the Rochlin-Ochanine theorem, which says that $16$...
- 6,933
12
votes
0
answers
231
views
Does cohomology ring determine a compact symmetric space?
Suppose that $M_1, M_2$ are compact connected symmetric spaces with isomorphic integer cohomology rings. Does it follow that $M_1$ is diffeomorphic to $M_2$?
The only result I am aware of is this ...
- 9,749
12
votes
0
answers
381
views
Looking for an invariant similar to algebraic K-theory
I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each small ...
- 40.2k
12
votes
0
answers
403
views
The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
- 7,549
12
votes
0
answers
493
views
The homotopy theory presented by a Waldhausen category
Waldhausen introduced his categories for the purposes of defining algebraic $K$-theory of suitable categories. From a modern perspective, it looks like he was really doing two things at once:
...
- 61.5k
12
votes
0
answers
101
views
How many upper sets in this decomposition of finite posets
Let $X$ be a finite poset.
If
$$X = X_1 \cup X_2$$
where $X_1$ and $X_2$ are strict upper sets, then a lot of properties of $X$ can be inferred from the smaller posets $X_1, X_2$ and $X_1\cap X_2$ (...
- 221
12
votes
0
answers
650
views
Understanding a certain algebraic set arising in Deep Learning
I'm not a professional geometer. Thanks in advance for your patience.
So, let $n$, $k$, $p_0,\ldots,p_{k}$ be positive integers. Let $X$ (resp. $Y$) be an $p_0$-by-$n$ (resp. an $p_{k}$-by-$n$) real ...
- 6,726
12
votes
0
answers
885
views
Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...
- 711
12
votes
0
answers
498
views
Does Lefschetz-type theorems imply ampleness?
Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties:
$D$ is ample.
(Positivity) For any $k$-dimensional ...
- 1,115
12
votes
0
answers
858
views
Conditions for the second homotopy group to be Abelian
What is the weakest set of assumptions on a pair of spaces $X\subset M$ for which the second homotopy group $\pi_2(M,X) $ is guaranteed to be Abelian?
Naively, I expected that Abelian $\pi_1(X)$ ...
- 393