Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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4 votes
1 answer
194 views

Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?

I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$. How about for multiple summands in the ...
3 votes
1 answer
250 views

non-simple local coefficient system on a fibration of classifying spaces

Long story short; I posted in MSE https://math.stackexchange.com/questions/2500745/local-system-of-coefficients-on-a-fibration-of-classyfing-spaces It is well known that if $G$ is a lie group ...
6 votes
2 answers
305 views

a comparison between LS and cohomological dimension

Let $X$ a simply connected elliptic space. Assume $\pi_\star(X)\otimes\Bbb{Q}$ is concentrated in odd degrees. So, we have $dim~\pi_\star(X)\otimes\Bbb{Q}=TC(X_\Bbb{Q})=catX_\Bbb{Q}$ (ie) the ...
24 votes
2 answers
1k views

Good functorial model for BG

There are several functorial constructions of the space BG for a topological group (meaning BG plus the universal G-bundle). First, there is the Milnor construction, treated in several textbooks. The ...
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 ...
6 votes
1 answer
684 views

Bialynicki-Birula Decomposition and moment polytopes/graphs

Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/...
5 votes
1 answer
577 views

Loop space of a Simplicial Abelian group

Let $X$ be a simplicial abelian group. Let $NX$ be its normalised chain complex denoted ...$\rightarrow NX_{K}$ $\rightarrow$ $NX_{k-1}$ $\rightarrow$... Now define a new chain complex $Y$ by ...
13 votes
1 answer
700 views

A question on connected sum of compact manifolds

Let $M$ be a compact orientable manifold which is homeomorphic to its connected sum with itself $M\# M$. Must $M$ be homeomorphic to a sphere? I will explain why I am interested (at the risk of being ...
3 votes
1 answer
230 views

Is a simply connected elliptic space rationally homotopy equivalent to a loop space or a suspension?

Let $X$ be an elliptic simply connected space. Is it rationally homotopy equivalent to the suspension of some connected space $Y$? If not, is it rationally homotopy equivalent to a loop space?
8 votes
1 answer
282 views

Is the homology of $\Omega^2\Sigma^2X$ free as a Gerstenhaber algebra?

Let $X$ be a connected space. According to Getzler BV-algebras and two-dimensional topologcial field theories , page 271, we have and isomorphism $ H_*(\Omega^2\Sigma^2X) \cong {\cal G}( \widetilde{H}...
6 votes
2 answers
424 views

Decribe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$

According to this MO post, there is two possible $S^2$ fibration over $S^2$. One is obviously $S^2\times S^2$, another one is the connected sum of two copies of $\mathbf {CP}^2$ with different ...
15 votes
0 answers
355 views

Cohomology with compact support for determinant varieties

I wonder if anyone knows anything about the cohomology with compact supports for determinantal varieties, such as the varieties of $m \times n$ matrices of full rank.
2 votes
1 answer
285 views

connectedness of fibers of torus-equivariant moment maps

Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or ...
2 votes
3 answers
2k views

Lifts across covering maps

Let $X,Y,Z$ be connected topological spaces, $f\colon X\to Y$ be a continuous map and $p\colon Z\to Y$ be a covering map. The problem is the existence of a continuous lift of $f$ across $p$. A ...
30 votes
2 answers
2k views

Does there exist any non-contractible manifold with fixed point property?

Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
6 votes
0 answers
169 views

Are 2d gauge anomalies determined by genus-one data?

Let $G$ be a (finite, say) group and $\alpha \in \mathrm{H}^3(\mathrm{B}G; \mathrm{U}(1))$ a 3-cohomology class. For each oriented 3-manifold $X^3$ equipped with a $G$-bundle $P : X \to \mathrm{B}G$, ...
23 votes
2 answers
810 views

Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...
0 votes
2 answers
216 views

If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?

Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...
13 votes
1 answer
552 views

Realizing symmetric groups by diffeomorphisms

Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects ...
1 vote
0 answers
109 views

Triangulation induces morphism of Cochain Complexes

Let $X$ be a topological space, $R$ a ring, $n \in \mathbb{N}$ natural. Let $S_n(X, R) = \bigoplus_{s: \Delta_n \to X} R$ where the $s: \Delta_n \to X$ are the singular n-simplices, therefore ...
8 votes
1 answer
3k views

Homotopy equivalence from contractibility of fiber

Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that $...
4 votes
2 answers
412 views

Do trivial homotopy groups imply existence of boundary preserving homotopies?

This is a cross-post from MSE. Let $N$ be a smooth $d$-dimensional connected orientable manifold which have the following property: For every smooth $d$-dimensional manifold $M$ with non-empty ...
14 votes
3 answers
2k views

Infinity-categories vs Kan complexes

It is known (cf. Lurie's book Higher Topos Theory for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka ...
14 votes
2 answers
1k views

Pullback and homology

Suppose I have two maps of topological spaces, $f:X\rightarrow B$ and $g:Y\rightarrow B$, such that $f$ induces a homology isomorphism and $g$ is a fibration and $B$ is connected. Is it true that the ...
2 votes
1 answer
149 views

Homological dimension of configurations spaces

Please feel free to delete or move it to somewhere. I just need a confirmation or a reference. Let $D_r(\mathbb{R}^l,S^n)=F(\mathbb{R}^l,r)_+\wedge_{\Sigma_r}(S^n)^{\wedge r}$ be the $r$-th stable ...
5 votes
1 answer
500 views

Attribution of theorem saying that inducing isomorphism on homology implies homotopy equivalence between H spaces that are CW complexes

Who was the first to prove this theorem and is there an "official" name for it? Let $\phi:X\rightarrow Y$ be a map of H-spaces that are also CW-complexes. Assume $\phi$ induces isomorphisms on ...
7 votes
3 answers
307 views

Eilenberg-Zilber-type theorem for good fiber products?

My question is: If $p\colon X \to B$, $q\colon Y \to B$ are proper submersions, is there a characterization of $H_*(X \times_B Y)$ in terms of $H_*(X)$, $H_*(Y)$, $H_*(B)$ that is simpler than the ...
6 votes
0 answers
135 views

How to decide a closed Top/PL manifold is a boundary?

For a closed smooth manifold, we can use the Stiefel-Whitney number of the manifold to decide it is boundary or not. For a closed topology or PL manifold, how to decide it is a boundary of compact ...
10 votes
2 answers
484 views

Copies of topological fundamental groups inside etale fundamental groups given by different embeddings of your field into $\mathbb{C}$

Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$. Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract ...
9 votes
1 answer
803 views

6-manifolds admitting SO(3) action with 2 orbit types

Let $M^6$ be a 6-dimensional smooth manifold, on which the group $G=SO(3)$ acts smoothly with 2 orbit types $SO(3)/SO(2)$ and $SO(3)$, such that the orbit space $X=M/SO(3)$ is a 3-ball $B^3$, whose ...
7 votes
3 answers
548 views

Stiefel-Whitney class of an orthogonal representation

Let $BG$ denote the classifying space of a finite group $G$. For which group cohomology classes $c\in H^2(G;\mathbb{Z}/2)$ does there exist a real vector bundle $E$ over $BG$ such that $w_2(E)=c$?
15 votes
1 answer
537 views

Is this generalization of Borsuk Ulam true? Roots of unity

Consider a continous map from $S^2$ to $C$. Is it true that there exists 3 points equially spaced on a great circle, $x_1,x_2,x_3$, such that if $w$ is the third root of unity, $f(x_1)+wf(x_2)+w^2f(...
10 votes
2 answers
625 views

Seeking very regular $\mathbb Q$-acyclic complexes

This question was raised from a project with Nati Linial and Yuval Peled We are seeking a $3$-dimensional simplicial complex $K$ on $12$ vertices with the following properties a) $K$ has a complete $...
9 votes
1 answer
4k views

General gluing theorem for adjunction spaces

Consider the following interesting theorem (7.5.7, p.294 in Topology and Groupoids by Ronald Brown): Gluing theorem for adjunction spaces: Suppose that we have the following commutative diagram of ...
-6 votes
1 answer
265 views

The Betti numbers of of $CP^n\sharp CP^n$ [closed]

I have known that $b_2(CP^n\sharp CP^n)=2$, however I have no idear how to prove this fact ! I appreciate any help for this simple question! Thank you!
9 votes
2 answers
414 views

"Lagrangian" subalgebra of cohomology, with respect to Poincare duality?

Let $M$ be a compact oriented $n$-manifold, and let $H^*(M)$ denote its cohomology ring with coefficients in $\mathbb{R}$. Let's say that a graded subalgebra $K^\bullet \subset H^\bullet(M)$ is a ...
17 votes
4 answers
2k views

Characteristic classes in generalized cohomology theories?

Hello, 'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (...
29 votes
2 answers
1k views

Quillen + construction for finite groups

Is there an example of two non isomorphic finite groups $G$ and $H$ such that $BG^{+}$ is homotopy equivalent to $BH^{+}$ ?
9 votes
1 answer
621 views

Must an inverse limit of simply connected groups be simply connected?

While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...
19 votes
2 answers
1k views

Is there a geometric interpretation for Reidemeister torsion?

Given a finite CW or simplicial decomposition of a space $X$ and a ring homomorphism $\varphi:\mathbb{Z}[\pi_1(X)]\to F$ for a field $F$, if the $\varphi$-twisted homology is trivial, then the ...
7 votes
0 answers
359 views

Obstructions for existence of a fiber wise covering space structure( A bundle of covering spaces)

Let $S^n \times \mathbb{T}^n$ be the trivial Torus bundle over $S^n$. Assume that we have a continuous fiber preserving map $\phi :TS^n \to S^n \times \mathbb{T}^n$ which restriction to each fiber ...
8 votes
2 answers
541 views

What is the fundamental group of $\mathcal O_{\mathbb P^n}(k)$ minus the zero section

Let $L^*$ be the total space of the line bundle $\mathcal{O}_{\mathbb{P}^n}(k)$ minus its zero section. How can one compute the fundamental group of $L^*$? For k = 0 the space $L^*$ is $\mathbb{P}^n ...
8 votes
0 answers
335 views

$C_2$-equivariant Betti realization of MGL

Let $MGL$ denote the motivic spectrum representing algebraic cobordism. Over $\mathop{Spec}(\mathbb{C})$ there is a Betti realization functor $\mathop{SH}(\mathbb{R}) \to \mathop{SH}$, which takes $...
8 votes
1 answer
200 views

Todd genus of symplectic $4$-manifolds a smooth invariant?

Suppose that $(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ are compact symplectic $4$-manifolds, that are (oriented) diffeomorphic. Is it true that the Todd genus ($\frac{1}{12} (c_{1}^{2} + c_{2})(M_{...
0 votes
1 answer
629 views

How to understand the simple closed curves in torus?

Let $\alpha$ and $\beta$ be two simple closed curves in $T^2$ that intersect each other in one point. We identify $\alpha$ with $(1,0)\in \mathbb{Z}^2$ and $\beta$ with $(0,1)\in\mathbb{Z}^2$. Let $(...
9 votes
0 answers
227 views

Chromatic Completion of Suspension Spectra and affine results

There is the Chromatic Convergence Theorem by Hopkins and Ravanel which states that the homotopy inverse limit of the chromatic tower of a finite spectra $X$ is $X$. Let's call any spectra with this ...
4 votes
0 answers
297 views

$\pi_0$ in arbitrary category of simplicial objects

Let $\mathcal C$ be a category (let it be pointed and cocomplete) such that the category of simplicial objects $s\mathcal C$ is a model category. In particular, I'm interested in two cases: $\mathcal ...
12 votes
2 answers
1k views

Universal covering of a 2-sphere without $n$ points

Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic ...
80 votes
2 answers
7k views

Vladimir Voevodsky's works

Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
6 votes
1 answer
187 views

What functions have the same persistence diagrams?

The panels in the figure below show, from left to right: a piecewise affine function with support equal to a bounded interval and an indication of its superlevel filtration; the corresponding ...

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