Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,255
questions
8
votes
0
answers
270
views
Combinatorial spin$^{\mathbf{C}}$ structures
Below is a brief introduction to spin$^{\mathbf{C}}$ structure that I took from Wikipedia. For more information, one should refer to https://en.wikipedia.org/wiki/Spin_structure#SpinC_structures.
A ...
- 875
4
votes
1
answer
293
views
Compute characteristic classes of principal bundle over closed surfaces
Let $G$ be a connected Lie group and $\Sigma$ a closed oriented surface. We know that principal $G$-bundles $P$ can be topologically classified by a characteristic class $c(P)\in H^2(\Sigma,\pi_1G)\...
- 201
7
votes
1
answer
610
views
Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:
By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ ...
- 312
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 ...
- 273
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 ...
- 1,219
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
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 ...
- 121
6
votes
1
answer
689
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/...
- 1,679
5
votes
1
answer
579
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 ...
- 371
7
votes
2
answers
716
views
Does there exist a Haken manifold where all its incompressible surfaces are non-separating?
Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A ...
- 3,571
13
votes
1
answer
706
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 ...
- 471
20
votes
2
answers
2k
views
What is the status of the 4-dimensional Smale Conjecture?
4-dimensional Smale conjecture claims the following:
The inclusion $SO(5)$ → $SDiff(S^4)$ is a homotopy equivalence.
or Does $Diff(S^4)$ have the homotopy-type of $O(5)$ ?.
The inclusion $SO(n + 1$)...
- 479
3
votes
1
answer
231
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?
- 121
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}...
- 1,935
15
votes
2
answers
1k
views
Must any continuous odd map from $\mathbb{S}^2$ to $\mathbb{R}$ have a path of zeros between antipodal points?
Let $f : S^2 \to \mathbb{R}$ be a continuous map such that $f(-x) = -f(x)$. Consider the set $Z = f^{-1}(0)$. Must $Z$ contain some path from some point to its antipode? Indeed, must $Z$ contain a ...
- 5,652
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.
- 151
6
votes
0
answers
170
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 $P : X \to \mathrm{B}G$, ...
- 53.1k
2
votes
1
answer
286
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 ...
- 1,679
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 ...
- 6,000
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 ...
- 3,571
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 ...
- 6,621
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 ...
- 1,619
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 ...
- 3,071
41
votes
12
answers
6k
views
Why is the definition of the higher homotopy groups the "right one"?
If someone asked me the question for the fundamental group, I would answer as follows:
The connection to classification of covering spaces.
The fundamental group of many spaces is an object of ...
16
votes
2
answers
1k
views
Teaching Steenrod Operations
I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...
- 161
13
votes
1
answer
416
views
Does Grayson/Quillen's "pre group completion" have a universal property?
Algebraic $K$-theory of a symmetric monoidal category $C$ is defined in two steps: first take geometric realization, then group complete: $K(C) = \Omega B |C|$.
In HAK II, Grayson (following Quillen)...
- 61.5k
5
votes
1
answer
502
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 ...
- 1,727
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 ...
- 1,749
3
votes
1
answer
131
views
Survey article on quasitoric manifolds
I am looking for a good overview on quasitoric manifolds. I have read Toric Topology by Buchstaber and Panov which was good but I was wondering if there is something that has more. Something like a (...
- 389
10
votes
2
answers
486
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 ...
- 10k
15
votes
1
answer
538
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(...
- 515
-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!
- 89
8
votes
1
answer
536
views
Question regarding the paper by Atiyah, Bott and Shapiro: alternative description of K-theory
In Atiyah, Bott, and Shapiro - Clifford modules (journal, MSN), the authors discuss the alternative description of K-theory in terms of sequences of vector bundles. I would like to understand the ...
- 9,150
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 ...
- 413
7
votes
3
answers
308
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 ...
- 1,569
8
votes
0
answers
337
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 $...
- 3,655
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_{...
- 6,933
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 ...
- 312
6
votes
2
answers
425
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 ...
- 1,610
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 ...
- 879
12
votes
1
answer
1k
views
Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms
I asked this question a while ago on MSE, got no answer, put a bounty on it, still got no answer, was advised to ask here instead, hesitated, forgot about the question for a while and now remembered ...
- 9,539
4
votes
0
answers
300
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 ...
- 365
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 $BH^{+}$ ?
- 1,619
5
votes
3
answers
845
views
Poincaré duality for Deligne (co)homology
My question is about the papers
Hélène Esnault and Eckart Viehweg, Deligne-Beı̆linson cohomology
Uwe Jannsen, Deligne homology, Hodge-D-conjecture, and motives
(both from the Beı̆linson's ...
- 181
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 ...
- 21.1k
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 ...
- 591
16
votes
2
answers
2k
views
Mathematical/Physical uses of $SO(8)$ and Spin(8) triality
Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation ...
- 10.3k
1
vote
0
answers
81
views
example of smoothing of quotient surface singularities with maximal Milnor number
In Wahl's paper "Smoothing of normal surface singularities", he shows that smoothing in the Artin component of a quotient surface singularity has the maximal Milnor number in the versal family. ...
- 31
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 ...
- 15.3k
0
votes
0
answers
115
views
Open subsets of the n-torus containing no nontrivial loops
Let $T^n$ denote the $n$-dimensional torus. Suppose there is an open subset $U\subset T^n$ not containing any nontrivial loop. Does this imply that the inclusion $U\hookrightarrow T^n$ is ...
user83520