Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,225
questions
3
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0
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79
views
A well pointed inclusion induces injection of pointed homotopy classes
Strom, 5.144 shows that there exists a short exact sequence of pointed sets
$$
0 \to [X,Y]\overset{\phi}\to \langle X,Y\rangle \to \pi_0(Y)\to 0
$$
where
$X$ is well-pointed, i.e. $*\to X$ is a ...
2
votes
1
answer
179
views
Cobordism/bordism group based on orbifolds with corners
We define a geometric homology group of a topological space $X$ as follows: the chain complex $C_{\bullet}$is freely generated by the maps $f$ from a compact oriented orbifold with corners $P$ to $X$, ...
0
votes
0
answers
69
views
induced homomorphism of adjoint action on fundamental group
Let $G$ be a non-connected compact Lie group and $Ad_g: G \rightarrow G, x \rightarrow gxg^{-1}$ be the adjoint action. Look at the induced homomorphism $ (Ad_g)_* : \pi_1 (G) \rightarrow \pi_1(G)$. ...
9
votes
2
answers
534
views
Generalize Wu formula to integral cohomology classes
For $\mathbb{Z}_2$ cohomology classes, we have a very useful Wu formula:
In $d$-dimensional manifold and for a $n$-cocycle in $x_n \in H^n(M^d; \mathbb{Z}_2)$, we have $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, ...
3
votes
0
answers
149
views
Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
15
votes
1
answer
513
views
What are the advantages of various "models" for the motivic stable homotopy category
People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...
15
votes
2
answers
899
views
What are the tangent $\infty$-categories to $\mathrm{Top}^\mathrm{op}$ and $E_\infty$-$\mathrm{Ring}^\mathrm{op}$?
Given an $\infty$-category $\mathcal{C}$ with finite limits and finite colimits, there are two ways to make it stable. One is to pass to the pointed objects and take the category of $\Omega$ objects $\...
12
votes
3
answers
949
views
Fixed point set of smooth circle action
Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
7
votes
5
answers
1k
views
The Hopf invariant is an isomorphism for $\pi_3 (S^2)$
Does any one have a reference to a proof that the Hopf invariant classifies the homotopy classes of maps from $\mathbb{S}^3$ to $\mathbb{S}^2$.
It is quite standard to find a proof that the Hopf ...
7
votes
0
answers
213
views
Notation: Why Ω for the based loop functor?
This is just a question about notation - probably useless, but it's always baffled me:
Why was $\Omega$ chosen to denote the based loop functor?
I once heard someone speculate: "It's because $\Omega$...
15
votes
0
answers
385
views
References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology
Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
15
votes
2
answers
913
views
Are strict $\infty$-categories localized at weak equivalences a full subcategory of weak $\infty$-categories?
One has a nice "folk" model structure on strict $\infty$-categories due to Yves Lafont, Francois Metayer and Krzysztof Worytkiewicz whose notion of weak equivalences seem to be the notion of weak ...
8
votes
0
answers
273
views
Geometric meaning of Aomoto complex
Generally, if we have any space $X$, then multiplication by any odd class $\eta$ makes $H^*(X)$ a complex (usually called Aomoto complex) $A_{\eta}$ because cup product is graded commutative. It is ...
5
votes
0
answers
135
views
Reference for model structure on non-positively graded cdga's over a field of characteristic 0
I was wondering if there were some references where the (projective?) model structure on $ cdga_k^{\leq 0}$, with $char(k) = 0$, and $deg(\partial) =1$ was explicitly described in detail, possibly ...
21
votes
0
answers
705
views
If $X\times Y$ is homotopy equivalent to a finite-dimensional CW Complex, are $X$ and $Y$ as well?
Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-...
25
votes
2
answers
1k
views
Why not $\mathit{KSO}$, $\mathit{KSpin}$, etc.?
If $X$ is a compact Hausdorff space, we can consider the Grothendieck ring of real vector bundles on $X$,
$\mathit{KO}^0(X)$, and this extends to a generalized cohomology theory represented by a ring ...
8
votes
1
answer
880
views
Algebraic fundamental group of a variety
I have a very explicit question. Consider a projective variety (a Fano 3-fold) in $\mathbb P^{10}$ defined by 3 quadrics and 32 cubic equations. I want to show that the algebraic fundamental group of ...
2
votes
1
answer
576
views
Topological spaces without higher homotopy and homology groups
Is there anything good in the class of objects with trivial higher homotopy and homology groups? Can it be described in some terms?
For example:
such $X$ that $\pi_{\gg 0}(X) = 0$ and $H_{\gg 0}(X,\...
8
votes
3
answers
621
views
Stiefel-Whitney total class with prescribed zeros
First things first, I am aware of the existence of this topic. It's related, but old and my question hasn't been discussed there. So I hope it's not wrong to start a new topic.
I'm currently ...
2
votes
1
answer
367
views
Homotopy groups of noncommutative spaces
In the approach to noncommutative geometry of Alain Connes any Hausdorff compact space $X$ is replaced by its algebra of complex valued continuous functions $C^0(X)$, and one regard general (that is, ...
11
votes
3
answers
1k
views
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
My question is
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
(My thoughts on this which might not be useful at all.) Since an ...
16
votes
3
answers
1k
views
SO(3) action on (simply connected) 6 manifold with discrete fixed point
If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
5
votes
1
answer
715
views
To derive or not to derive, that is the question
What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" ...
4
votes
3
answers
369
views
Can one calculate possible mapping degrees from a connected-sum to another manifold?
Let $D(M,N)$ be the set of all possible degrees of maps from $M$ to $N$, $M_1\#M_2$ the connected sum of $M_1$ and $M_2$.
Can $D(M_1\#M_2,N)$ be calculated in terms of $D(M_1,N)$ and $D(M_2,N)$?
...
9
votes
1
answer
472
views
What does positivity of the first Pontryagin number of a vector bundle tell us?
Some context:
In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\...
15
votes
2
answers
2k
views
Every 4-manifold has a $\operatorname{Spin}^c$ Structure
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}$I'm having trouble understanding the proof given in Morgan's The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-...
7
votes
1
answer
617
views
Computing the equivariant cohomology of a specific $(\mathbb{Z}/2\mathbb{Z})^2$-space
In the paper On the Castelnuovo-Mumford regularity of the cohomology ring of a group, Symonds describes the following space.
Let $G = (\mathbb{Z}/2\mathbb{Z})^2 = \{1,a,b,ab\}$ be an elementary ...
5
votes
1
answer
334
views
Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points?
It is well known that the set of all polygons with consecutive side lengths $l_1, \dots, l_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming,...
3
votes
1
answer
272
views
What is the relation between cobar duality and Feynman transform
If $O$ is a cyclic operad, it can be regared as a modular operad $P$ with $P(g,n)=0$, for $g >0$. So we have cobar dual $BO$ and Feynman transform $FP$(with trivial cocycle). Is there any ...
10
votes
2
answers
479
views
Complex varieties with non-torsion homotopy groups
Is there some kind of classification of (connected) smooth complex varieties such that every homotopy group of the manifold of complex points is torsion-free? Any reference on this topic will be most ...
20
votes
1
answer
1k
views
Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space
Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence
$$H^*(BG;K^*) \implies K^*(BG)$$
connecting $H^*BG$, which generally contains torsion, with $K^*BG \cong \...
15
votes
0
answers
624
views
Dijkgraaf-Witten topological invariant
We know that given a finite group $G$ and its group 4-cohomology class $w \in H^4[G;U(1)]$, we can obtain a DW topological invariant $Z_{G,w}(M^4)$ as the partition function of the DW theory on a ...
5
votes
1
answer
2k
views
t-Stochastic Neighbor Embedding vs Topological Data Analysis
The shortest form of this question is:
How much TDA can be done with tSNE?
Specifically, I'm referring to the application of TDA to clustering data, so, think along the lines of Ayasdi's ...
6
votes
0
answers
479
views
A property of slant product in group cohomology
Recently I have encountered a problem concerning the property of slant product in group cohomology. The problem is as follows:
Consider a finite group G (can have anti-unitary operations). And there ...
40
votes
0
answers
1k
views
Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?
Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
4
votes
1
answer
425
views
Representing homology classes in a Heegaard diagram
Given a Heegaard diagram $(\Sigma, \alpha, \beta)$ we obtain a compact 3-manifold $M$ together with a handle decomposition where the $\alpha$ curves are the belt spheres of the 1-handles and the $\...
5
votes
1
answer
306
views
Expression of morphisms in motivic homotopy categories in terms of Nisnevich cohomology?
For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules ...
40
votes
1
answer
4k
views
Proof that a local fibration is a fibration, in May
I was reading "A Concise Course in Algebraic Topology" by J.P.May (page 52) and found the proof of the following theorem incomprehensible:
Let $p:E\rightarrow B$ be a map and let $\mathcal{O} $ be a ...
2
votes
0
answers
394
views
Terminology for "global sections" when sheaf is valued in general category
Let $\mathcal F$ be a sheaf (say on a topological space $X$) valued in some category $\mathcal C$.
What do we call $\mathcal F(X)$?
When $\mathcal C$ is some vaguely linear category (e.g. the ...
3
votes
1
answer
155
views
Cyclic polytopes whose boundary is a flag complex
A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ ...
2
votes
0
answers
69
views
Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
1
vote
1
answer
280
views
Classifying spaces of finitely presented groups with torsion elements
Suppose $\Gamma$ is a finitely presented group that has a torsion element. Can the classifying space $K(\Gamma,1)$ be homotopic to a finite-dimensional manifold?
If yes, what is the simplest example?
6
votes
2
answers
320
views
Where to find the proof that these two version of simplicial homotopy are equivalent?
Let $f,g: X_{\bullet}\to Y_{\bullet}$ be two simplicial maps between simplicial sets. We say $f$ and $g$ are (strictly) simplicial homotopic if there exists a simplicial map
$H: X_{\bullet}\times I_{\...
2
votes
0
answers
309
views
The subtlety with (an algebraic phrasing of) the Whitehead conjecture?
The Whitehead conjecture states that if $X$ is a $2$-dimensional aspherical simplicial complex and $Y \subset X$ is a connected sub-complex then $Y$ is aspherical. This can be re-phrased in terms of ...
8
votes
2
answers
377
views
Maps to $K(\pi,1)$ spaces from manifolds with $S^1$-action
Suppose $M$ is a connected smooth manifold with a smooth $S^1$-action that fixes a point in $M$. Let $X$ be a $K(\pi,1)$-space and let $\varphi: M\to X$ be a continuous map.
Question. How to prove ...
1
vote
1
answer
112
views
Is singular barycentric subdivision injective?
This question has been asked on mathstackexchange without any answers.
Let us note $\Delta_p(X)$ the $p$-singular chains on a topological space $X$. We have a well-known barycentric subdivision
$$b:...
3
votes
1
answer
586
views
Gottsche Nakajima Yoshioka define a weird slant product
In their article Instanton counting and Donaldson invariants the authors define the slant product for $\beta \in H_i(X)$ (where $X$ is a manifold) as following.
Let $P \to X$ and SO(3) bundle and $M(...
8
votes
1
answer
330
views
Lifting Strict Comonoids and Comodules to Quasicategories
$\newcommand{\M}{\mathcal{M}}$
Suppose I have a monoidal simplicial model category in which every object is cofibrant $(\M,\otimes,\mathbb{1})$ and I want to look at its underlying monoidal ...
2
votes
2
answers
305
views
How is the equivariant cohomology of a space related to the cohomology of the corresponding associated bundle
Let $X$ be a manifold with a left $G$-action, and let $\Sigma$ be a Riemann surface. How is the equivariant cohomology $H^*_G(X)$ of $X$ related to the de Rham cohomology of the associated bundle $H^*(...
6
votes
0
answers
182
views
Deformation theory over F_p
Lurie proves that formal $E_\infty$ moduli problems over a field $k$ are equivalent to augmented $E_\infty$-algebras.
Is there a reasonably small model for this when $k = \mathbb{F}_p$?
Or maybe we ...