Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,255
questions
7
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654
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FIltered colimits of truncated objects in $\infty$-topoi
The bare question:
Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT 6....
- 3,386
7
votes
1
answer
797
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Which Abelian Group sequences arise as the Homology of Embedded CW Complexes?
Background
Let $\mathcal{A} = \lbrace A_0, \ldots, A_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X_\mathcal{A}$ may be ...
- 15.4k
7
votes
1
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1k
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G-equivariant Whitehead's Theorem
Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume ...
- 8,394
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1
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579
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How does this geometric description of the structure of PSL(2, Z) actually work?
There is a beautiful way to see that the congruence subgroup $\Gamma(2)$ is free on two generators: the action of $\Gamma(2)$ on $\mathbb{H}$ is free and properly discontinuous, and there is a modular ...
- 115k
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1
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374
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Is anything known about de Rham $K(\pi,1)$'s?
Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the ...
- 975
7
votes
2
answers
446
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Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is ...
- 374
7
votes
2
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452
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Injectivity of the cohomology map induced by some projection map
Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...
- 125
7
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1
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293
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Is it possible to separate two linked (geometric) circles in $\Bbb R^3$ by a set homeomorphic to the 2-sphere (with arbitrarily “bad” homeomorphism)?
$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, ...
7
votes
1
answer
384
views
Does a Gysin map depend on the choice of Thom class?
Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda_N$ and a Thom ...
user484289
7
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2
answers
745
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Inclusion–exclusion principle for the compactly supported Euler characteristic
If $M$ and $N$ are sufficiently nice subspaces of some topological space $X$ then their Euler characteristics obey an inclusion-exclusion principle:
\begin{equation}
\chi(M) + \chi(N) = \chi(M\cup N) +...
- 133
7
votes
1
answer
440
views
The group structure on $[X,S^n]$ induced by the framed bordism
I'm concerned about the group structure on $[X,S^n]$, i.e. the set of homotopy classes of continuous maps from $X$ to $S^n$.
On the one hand, $[X,Y]$ has a group structure that is natural with respect ...
- 541
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votes
1
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330
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Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?
The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but ...
7
votes
1
answer
328
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Intersection form of surface bundle over surface
Let $\Sigma_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle over surface: $\Sigma_g \to M^4 \to \Sigma_h$. $\Sigma_g$ is the fiber and $\Sigma_h$ is the base space.
My question: ...
- 4,716
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403
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Visualize (co)sketeton of a simplicial set (geometrical intuition)
I want to understand if there is an intuition approchable with
most possible 'elementary geometrical' knowledge for
$n$-(co)skeleta of simplicial sets?
Formally sketleton & coskeleton functions ...
- 6,000
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279
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Is $Tor_A(k,k)$ a bicommutative Hopf algebra?
Let $A$ be a commutative (or graded commutative) algebra over a field $k.$ In some sources, such as Mcleary's book on spectral sequences, Corollary 7.12, pg. 248, it is claimed that $\text{Tor}_A(k,k)$...
7
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1
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Classifying space of semidirect product of groups
Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...
- 2,525
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votes
1
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179
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If a loopspace admits space-level power operations, is is a higher loopspace?
Let $G$ be a group. Suppose that the map $G \to G$, $g \mapsto g^r$ is a group homomorphism for every $r \in \mathbb N$. Then $G$ is abelian. Is this also true homotopy-theoretically?
(In the ...
- 61.5k
7
votes
2
answers
292
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Homotopy domination of a wedge of two polyhedra
The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$.
Question: Suppose that $...
- 1,172
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votes
2
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981
views
Topology of the set of Nash equilibria of a normal form game
Consider a normal form game with $n$ players (and finitely many options per player) defined by finite option sets $A_1,\ldots,A_n$ and payoff matrices $u_1,\ldots,u_n: \prod_{j=1}^n A_j \to \mathbb{R}$...
- 30.1k
7
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1
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417
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Correspondence between persistence module and graded module over $R[t]$
In the paper "Computing Persistent Homology" by Zomorodian and Carlsson, it is stated as Theorem 3.1 that:
The correspondence $\alpha$ defines an equivalence of categories between the category of ...
- 1,219
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votes
3
answers
898
views
Lifting symmetries to the universal cover
If $X$ is a connected topological space with universal cover $p: \tilde{X} \to X$, I believe any homeomorphism $f : X \to X$ can be 'lifted' to a homeomorphism $\tilde{f} : \tilde{X} \to \tilde{X}$. ...
- 21.5k
7
votes
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405
views
classifying maps of Whitney sums of vector bundles
For an $n$-dimensional vector bundle $\xi$ with structure group $G\leq O(n)$ over a $CW$-complex $B$, we have a classifying map up to homotopy
$$
f(\xi): B\longrightarrow BG,
$$
$f(\xi)\in [B;BG]$, ...
- 519
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votes
1
answer
2k
views
Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles
As explained in:
Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds
principal $SU(2)$ bundles $P_{SU(2)}$ over a four-dimensional manifold $M$ are classified by their ...
- 3,064
7
votes
3
answers
2k
views
Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the Dold-...
- 36.3k
7
votes
1
answer
637
views
fiber bundle and free action
In Spanier's book " Algebraic topology" a fiber bundle is defined as follows:
A fiber bundle $\xi=(E,B,F,p)$ consists of a total space $E$, a base space $B$ and a fiber $F$ and a bundle projection $p:...
- 119
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votes
3
answers
431
views
Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?
Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces (...
- 10k
7
votes
1
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494
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Coefficients of real k-theory with coefficients
Question: Calculate the group $ \pi_{8k+2}(KO \wedge M\mathbb Z/l\mathbb Z) $.
Here $KO$ denotes the real k-theory spectrum and $M\mathbb Z/l\mathbb Z $ denotes a Moore Spectrum associated to the ...
- 347
7
votes
3
answers
885
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A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
- 13.2k
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votes
2
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1k
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The Norm Map in (group) cohomology via classifying spaces
The well-known transfer map in group (co)homology can be defined with only homological algebra, or with algebraic topology via classifying spaces (group cohomology of $G$ is isomorphic to ordinary ...
- 17.2k
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votes
2
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518
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Integrality of the canonical trace and topology
Let $G$ be a discrete group and consider the reduced group C* algebra $C_r^\ast(G)$, viewed as an algebra of bounded operators on $\ell^2(G)$ by the regular representation. The canonical trace on $...
- 28.8k
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votes
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1k
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homotopy pushout of spaces homotopic to finite CW complexes
Does anyone know a reference for the fact that a homotopy pushout (double mapping cylinder) of spaces which are homotopy equivalent to finite CW complexes is also homotopy equivalent to a finite CW ...
- 71
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1
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2k
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Details of Perelman's example about soul of Alexandrov space
Reading Perelman's preprint(1991) Alexandrov space II now. Got confused about the last section 6.4, which contains an example which indicate that the statement ".... manifold is diffeomorphic to the ...
- 1,101
7
votes
2
answers
397
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
7
votes
1
answer
802
views
Spectra and localizations of the category of topological spaces
Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces
using some kind of localization combined with other categorical ...
- 36.3k
7
votes
1
answer
578
views
Whitehead products on manifolds
What are some good examples of simply connected manifolds with interesting Whitehead Lie algebras over R? Most of the manifolds that one thinks about if one is pretty naive are not so interesting--- ...
- 3,656
7
votes
2
answers
536
views
(Co-) Homology associated to Waldhausen K-Theory
Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
- 1,253
7
votes
1
answer
553
views
Homotopy orbit spaces of representation spheres
Let $G$ be a finite group and $V$ be finite-dimensional real representation of $G$. Write $S^V$ for the one-point compactification of $V$, with induced $G$-action, viewed as a pointed space, and ...
- 24.9k
7
votes
1
answer
174
views
Generalisation of Hirsch formula for the associativity of Steenrod's higher $\cup_2$ product with $\cup_1$ and cup products
For $f$, $g$ and $h$ cochains, the Hirsch formula is given as
$$ (f\cup g)\cup_1 h=f\cup (g\cup_1 h)+(-1)^{q(r-1)}(f\cup_1 h)\cup g.$$
Is there a more general formula that relates the associativity of ...
- 71
7
votes
1
answer
418
views
Equivariant perverse sheaves and orbit stratification
Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$.
The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
- 3,024
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votes
2
answers
878
views
Is there a "spectral exterior algebra" construction in higher algebra?
Given a ring spectrum $R$ and an $R$-module $E$, we have the spectral symmetric algebra $\mathrm{Sym}_R(E)$ of $E$ over $R$, defined by
$$
\begin{align*}
\mathrm{Sym}_R(E) &\overset{\mathrm{def}}{=...
- 10.7k
7
votes
1
answer
123
views
Computing the homotopy groups $\pi_n(G,G_q)$
Let $G_q$ be the space of degree one maps $S^{q-1}\rightarrow S^{q-1}$ and $G=lim G_q$ under suspensions $G_q\subset G_{q+1}$. What are the known computations of $\pi_n(G,G_q)$?
I found that $\pi_3(G,...
- 71
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votes
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answer
411
views
Is there a topological interpretation of a module over $\Omega_{PL}(X)$?
Associated to a DGCA (differential graded commutative algebra) $A$, we can associate to it the category of modules over $A$. Hence, for a space $X$ we can consider the category of modules over $\...
- 5,201
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votes
1
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381
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A set theoretic question arising from trying to understand a sheaf cohomology question
I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
- 37.4k
7
votes
1
answer
297
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Homotopy in $X$ and homology in $X \times I$
Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f_1,f_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f_1$ and $...
- 5,319
7
votes
1
answer
452
views
Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair
Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...
- 537
7
votes
1
answer
436
views
Inequality number of facets simplicial complex
In a recent preprint, Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies a version of ...
- 7,200
7
votes
1
answer
922
views
Map Lifting lemma and Etale fundamental group
In algebraic topology, we have the map lifting lemma which says that given a covering space $p:(\tilde{X},\tilde{x})\rightarrow (X,x)$ and a map $f:(Y,y)\rightarrow (X,x)$ with $Y$ connected and ...
- 225
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
7
votes
2
answers
438
views
multiple zeta values and knots invariants
I have heard several times that MZV appear in the context of knot invariants and deformation quantisation. Could anyone explain how and give some references?
- 71
7
votes
1
answer
831
views
Schematization of a topological space
I wanted to understand or at least to know if what follows make sense.
Given a connected toplogical space $X$, I want to associate a scheme. In the following way.
For a space $X$ and $A(X)$ the ...
- 235