Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
2,259
questions with no upvoted or accepted answers
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de Rham cohomology with twisted coefficients and group cohomology
I'm trying to follow Goldman's paper 'The symplectic nature of fundamental groups of surfaces'. I'm having trouble understanding the following isomorphism: let's take a principal $G$-bundle $P \to M$ ...
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190
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mod $p$ homology module of unordered configuration spaces of the projective plane
Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
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344
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Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices
Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...
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212
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Reference request: Atiyah-Segal completion on spectrum level
It seems like the Atiyah Segal completion theorem for the two element group
$G = \mathbb Z_2$ and one-point space $X=\{ * \}$ with trivial G action yields a statement about the underlying spectra as ...
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251
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Topological localization of (infinite) inverse limits
The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
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394
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kernel of the mod $2$ Bockstein on the first cohomology group
Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
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507
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A question about cofibrations
Let $(X, A)$ be a cofibration, with $X$ compactly generated. This is equivalent to the fact that $A$ is a NDR of $X$, i.e., there exist two functions $\phi \colon X \rightarrow I$ e $H \colon X \times ...
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173
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Associated graded of double Koszul dual
Let $k$ be a field, and let $A$ be a graded, connected, augmented, locally finite $k$-algebra. If $\Omega^* A$ denotes the cobar complex of $A$ (i.e., the dual $Hom_k(B_*(A), k)$ of the bar complex ...
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633
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Is there an obstruction which classifies "quasi-isomorphism but not chain equivalence"?
Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...
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304
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Two different Thom diagonals in recent literature?
Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...
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405
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algebraic structure of Integral Steenrod squares
It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations
$$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$
In the case where $a$ is odd, one can define an ...
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175
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Consequences of the Euler characteristic vanishing mod p
Let $M$ be a smooth compact connected manifold. If $\chi(M)=0$, the unit tangent bundle $S(TM) \rightarrow M$ has a section. Is there anything that can be said if $\chi(M)$ is congruent to $0$ mod $p$ ...
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263
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Do there exist "non-algebraic tensor products" for "algebraic" triangulated categories?
Let us call a triangulated category algebraic if it admits a differential graded enhancement (i.e., an enrichment in complexes of abelian groups). Certainly, there is a notion of a tensor product on ...
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Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres
The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...
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Mapping class group action on fundamental group of punctured elliptic curves
Let $(\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}$ be the moduli stack of elliptic curves over $\overline{\mathbb{Q}}$. By Oda, we know that its etale fundamental group is $\widehat{SL_2(\mathbb{Z})}$.
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175
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Manifold approximations to $BO(3)$
We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$
Similarly $BO(2)$ can be approximated by closed, ...
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181
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$\eta$ invariants of Pin+ manifolds $\mathbb{RP}^{8k}$
In general, I am curious about the 'quantization' of $\eta$-invariants on Pin+ manifold, i.e., what is the 'minimal unit' of $\eta$-invariants on a manifold with certain choice of Pin+ structure.
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391
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Does the fat realization of simplicial spaces commute with finite limits up to homotopy?
I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, ...
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536
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Thom Class of tensor bundles
Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
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Reference: K-theory of real Grassmann manifolds
Was the complex K-theory of the unoriented real Grassmann manifold $G_{\mathbb{R}}(k,n)$ computed somewhere, at least for certain values of $k,n$?
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255
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May's infinite loop machine for Friedlander's result for Adams conjecture
Eric M. Friedlander in the paper The infinite loop Adams conjecture via Classification Theorem for $\Gamma$-spaces proved the infinite loop Adams conjecture using techniques involved $\Gamma$-space.
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479
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manifold branched covering space for orbifolds
An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of ...
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Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?
We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a non-...
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399
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Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups
I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:
(The proof is not finished yet but I am very confused by now.)
...
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776
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Commutation of simplicial homotopy colimits and homotopy products in spaces
Edit: The claim below is wrong, as explained in the comments,
because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...
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170
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From 3-framings on $\Sigma$ to $\mathrm{Spin}^c$-structures on $\mathrm{Loc}_G(\Sigma)$?
Here is my question, below that some motivation:
For $G$ a compact abelian Lie group and $\Sigma$ a surface, with $M_G = \mathrm{Loc}_G(\Sigma)$ denoting the space of flat $G$-connections on $\Sigma$ ...
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736
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Homotopy type of complex algebraic varieties
In his 1974 ICM adress "Poids dans la cohomologie des variétés algébriques", Pierre Deligne explains that any finite polyhedron has the same homotopy type as a complex algebraic variety (section 6.).
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250
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When is the projection of an induced fibration trivial on cohomology?
Let $p: E\to B$ be a fibration, and let $f: A\to B$ be a continuous map. In my applications, $E$ and $B$ are finite complexes, but $A$ need not be. Form the pullback
$$
\begin{array}{ccc} W & \to &...
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Induction map in equivariant K-theory
Let X be a space with $Z/2$ action. There is a map from $K(X)$ to the equivariant K-group $K_{Z_{2}}(X)$, which is called "the induction map". (It is a standard operation in equivariant stable ...
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Explicit expression of WZ term for orthogonal groups
Consider the Wess Zumino term on the the space $W=I\times D$, where D is a two dimensional disk disk and $I$ is an interval, $[0,1]$, say, i.e.,
$$
\int_{I\times D} \langle(u^{-1} \, du)^3\rangle
$$
...
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182
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Torsion in Whitehead group
Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
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389
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bar-cobar or cobar-bar
What is the standard or best reference for the adjointnes of bar and cobar constructions?
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191
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Central Extension of Continuous Loop Group
For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the ...
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374
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Kuranishi map, group cohomology and the bar complex
Let $\pi$ be a group, $G$ a compact lie group with lie algebra $g$, $A:\pi\to G$ a representation which composes with the adjoint map to give $g$ a $\pi$-module structure. I want to construct a ...
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177
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An explicit description of injective fibrations
If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would ...
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512
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homotopy pullbacks of tensor product chain complexes (towards Kunneth formula in diff cohomology)
I have editted this question from the previous version which did not obtain much attention.
Suppose I have two diagrams of chain complexes:
$A^* \rightarrow C^* \leftarrow B^*$
$\tilde{A}^* \...
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387
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Why the $M$ for Thom spaces?
I've heard $E$ is for entire space, $B$ is for base space, so what is $M$ for?
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428
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Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem
I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...
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360
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A question about a blue fan and a red fan and their common refinement
Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $...
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Inner product on Hochschild homology in 2d TCFTs
This should be an easy question for some people. Take a compact $A(\infty)$ algebra with a cyclically symmetric non-degenerate inner product. In Kontsevich and Soibelman's article "Notes on $A(\infty)$...
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106
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Explicit representatives for Borel cohomology classes of a compact Lie group?
I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) ...
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139
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Applications of $RO(G)$-graded computations outside of equivariant homotopy theory
While writing a grant proposal I faced a problem of justification my area of interest to a broader audience. So I thought it would be nice to ask it here:
What are applications/impact of computations ...
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142
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Extending topological vector bundles and obstruction theory
This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
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224
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Reference to a definition of a graph homology
Let $G$ be a graph, and define $C_k$ to be the free abelian group on the homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge ...
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165
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How exactly does the Kreck-Stolz description of elliptic homology match the one by Totaro?
In
Kreck, Matthias; Stolz, Stephan, $\mathbf H\mathbf P^2$-bundles and elliptic homology, Acta Math. 171, No. 2, 231-261 (1993). ZBL0851.55007.
the $n$th elliptic homology group of a space $X$ is ...
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199
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"Inclusion" between higher categories of framed bordisms?
Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds.
It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences.
If $n$ is large enough, ...
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127
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Mapping space between $n$-groupoids is an $n$-groupoid
Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where
$$
\underline{\mathrm{Hom}}(K,L)...
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583
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How to learn homotopy theory
I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy ...
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117
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K-homology fundamental class for singular varieties?
Given a smooth $\text{Spin}^c$ compact manifold without boundary $M$, a suitable normalization of the Dirac operator defines the fundamental class of $M$ in Kasparov's $KK(\mathbb{C}, C^0(M))$. This ...
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86
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Group structure on cohomology with coefficients in a spectral 2-type
Let $E$ be a spectrum having exactly two non-trivial homotopy groups, $\pi_k(E)=G$ and $\pi_j(E)=G'$ for $j>k\geq 0$, and having $k$-invariant $\alpha\colon\Sigma^{k}HG\to\Sigma^jHG'$. Also assume ...