Questions tagged [at.algebraic-topology]

Homotopy theory, homological algebra, algebraic treatments of manifolds.

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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 ...
fosco's user avatar
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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 $X$, ...
Hao Yu's user avatar
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0 answers
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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)$. ...
Xiaoyang Chen's user avatar
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 $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, ...
Xiao-Gang Wen's user avatar
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 ...
Guangbo Xu's user avatar
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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 ...
Mikhail Bondarko's user avatar
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 $\...
Tim Campion's user avatar
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12 votes
3 answers
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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 ...
Jens Reinhold's user avatar
7 votes
5 answers
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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 ...
Jean Van Schaftingen's user avatar
7 votes
0 answers
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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 $\Omega$...
user316092's user avatar
15 votes
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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 ...
Mikhail Skopenkov's user avatar
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 ...
Simon Henry's user avatar
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8 votes
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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 ...
Denis T's user avatar
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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 ...
Karl's user avatar
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21 votes
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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-...
David Sykes's user avatar
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 ...
Arun Debray's user avatar
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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 ...
abc's user avatar
  • 81
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,\...
Q. Q.'s user avatar
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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 ...
R. Alexandre's user avatar
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, ...
user avatar
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 ...
No One's user avatar
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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 ...
Yuhang Liu's user avatar
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" ...
Artur Jackson's user avatar
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 $D(M_2,N)$? ...
Invy S.'s user avatar
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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 $\...
Brian Klatt's user avatar
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-...
jdk3264's user avatar
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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 ...
Najib Idrissi's user avatar
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,...
Priyavrat Deshpande's user avatar
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 ...
Hao Yu's user avatar
  • 31
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 ...
user43198's user avatar
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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 \...
jdc's user avatar
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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 ...
Xiao-Gang Wen's user avatar
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 ...
Alex R.'s user avatar
  • 4,902
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 ...
Xu Yang's user avatar
  • 123
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 ...
Tim Campion's user avatar
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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 $\...
user101010's user avatar
  • 5,319
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 ...
Mikhail Bondarko's user avatar
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 ...
Gheehyun Nahm's user avatar
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 ...
John Pardon's user avatar
  • 18.3k
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 $\partial C(n, d)$ ...
Priyavrat Deshpande's user avatar
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 (...
annie marie cœur's user avatar
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?
aglearner's user avatar
  • 14k
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_{\...
Zhaoting Wei's user avatar
  • 8,657
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 ...
HeadingWhiteways's user avatar
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 ...
aglearner's user avatar
  • 14k
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:...
C. Dubussy's user avatar
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(...
Marion's user avatar
  • 577
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 ...
Jonathan Beardsley's user avatar
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^*(...
Mtheorist's user avatar
  • 1,135
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 ...
Vladimir Baranovsky's user avatar

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