Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
2,258
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Uniqueness of complex topological $K$-theory as an $S$-algebra
This might be well-known or trivial, but I could not figure out how to fill in the details: For an $S$-algebra $K$ denote its associated multiplicative cohomology theory by $h^*_K$. Suppose that I ...
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Why is the definition of a Dwyer map so particular?
Recall that a functor $F : A \to B$ is said to be a Dwyer map if the following two conditions are satisfied:
$F$ is a sieve inclusion.
There exists a factorization of $F$ as $A \to W \to B$ where $W ...
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Outer and inner automorphism of $\mathrm{Pin}$ groups
$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...
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Relation between Bott-Samelson theorem and James reduced product
I asked this question on the homotopy theory chat, but I haven't got any answer - thus I decided to post it as a question here.
The question is rather historical. Let $X$ be a connected topological ...
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(Co)homology of a directed space with coefficients in a commutative monoid
This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).
If I have a ...
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Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
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How does the topology of minimal surfaces depend on the radius?
Let $M^n \subset \mathbf{R}^{n+k}$ be a smooth, properly embedded minimal surface, with boundary $\partial M$. The convex hull property states that $M$ is contained inside the convex hull of its ...
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What do we know about the homotopy of $\mathrm{Q}S^0\wedge\mathrm{Q}S^0$?
The homotopy groups of $\mathrm{Q}S^0$ are the stable homotopy groups of spheres, of which we know the first 81 exactly, and the first 90 up to some uncertainties. A table is given in Wikipedia:
Do ...
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What do we know about $\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}$ and the spectral DM Stack $\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$?
These days I've been trying to wrap my head around the current proposed approaches to algebraic geometry over the elusive "field with one element", one of whose main objects of interest is ...
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Computation of partial configuration spaces ala Cohen
Let $F_i$ denote the subset of $\mathbb{R}^n$ consisting of tuples $(x_1,\dots,x_n)$ where there are less than or equal to $i$ unique entries. Is there a computation of the homology of $\mathbb{R}^n - ...
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329
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CW complex vs analytic manifold vs variety
I am looking to gain some intuition into the passage (or obstruction thereof) between different categories of objects one encounters in geometry and topology. To oversimplify things a bit, the ...
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Homotopy, contraction mapping and the inverse function theorem on Banach spaces
We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
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Construction of equivariant Steenrod algebra
I am reading through the calculations in Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" and I've got a small problem in understanding the ...
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Spectral sequence from a stratification by closed subvarieties
I am looking for a reference for the following result: If $X$ is an algebraic variety and
$$X = T_n \supset T_{n-1} \supset \cdots \supset T_{-1} = \varnothing$$
is a stratification (edit: filtration) ...
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Negative cyclic homology of the group algebra of discrete groups
I am looking for a reference for the calculation of the negative cyclic homology of the group algebra $\mathbb{K}[\Gamma]$ of a discrete group $\Gamma$ over a field $\mathbb{K}$ of characteristic 0. (...
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How to compute the class defined by intersection with a square?
$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n+k)$ (of course, one can do also for $\Gr(k,\infty)$) be the complex Grassmannian of $k$-planes in $n+k$-dimensional linear space.
It is well-known that ...
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Are there alternate descriptions of `elementary cobordisms'?
Let $M^d$, $N^d$ be cobordant $d$-manifolds. Then $M^d \sqcup \bar{N}^d = \partial W^{d+1}$ for some $(d+1)$-manifold $W$. This cobordism can be implemented via an elementary set of 'moves' called ...
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What is the image of the diagonal map on the cohomology of Lie groups
Consider a simple Lie group $G$ and its mod $p$ cohomology $H^*(G, \mathbb{Z}_p)$.
A good reference is the book
Mimura, Mamoru; Toda, Hirosi, Topology of Lie groups, I and II. Transl. from the Jap. by ...
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Minimal sum of Betti numbers of Kähler manifold with trivial canonical bundle
Let $M$ be a closed Kähler manifold of real dimension $2n$. Suppose the canonical bundle of $M$ is holomorphically trivial.
Is it true that $\sum_{i=0}^{2n} b_i(M)=n+3\implies n=1$?
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How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
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Equivariant splitting of loop space of a suspension
It is well known, e.g. by Cohen's "A model for the free loop space of a suspension", that there is a stable splitting of the free loop space $\mathcal{L}
\Sigma X $of the suspension $\...
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Galois action on torsion in homotopy groups not induced by homotopy equivalences
Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...
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Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
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chromatic minimal cell structures
If $X$ is a finite $p$-local spectrum, then the minimal number of cells needed to construct $X$ is exactly $\dim_{\mathbb F_p} H_\ast(X,\mathbb F_p)$. Is there an analogous result in the $K(n)$-local ...
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The homotopy type of the simplicial space obtained by free adding degeneracies to a semi-simplicial space
Let $\text{sTop},\text{ssTop}$ denote the categories of simplicial, semi-simplicial spaces respectively. There is a functor $E:\text{ssTop}\rightarrow \text{sTop}$ that is left adjoint to the ...
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240
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Finite CDGA model for a compact manifold
Is it true that a compact smooth manifold always has a finite-dimension commutative dg algebra model?
Same question can be asked about compact CW complexes.
More generally, is it true that a CW ...
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Where can I find W. Browder's thesis
I've been looking for W. Browder's thesis Homology of loop spaces for a while now, and I really found nothing except for articles and book having it in their bibliography. Does someone know if it can ...
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Reference request: Name or use of this group of diffeomorphisms of the disc
Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...
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Is this a stack?
A continuous map $f:X\to Y$ and a vector bundle $E\to X$ seem to give rise to a presheaf of groupoids on $Y$ along the following lines. For an open $U\subseteq Y$, each section of $f$ over $U$ (i. e. ...
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Bousfield-Kan and Generalized Eilenberg-Moore spectral sequences
Building on the work of Anderson and Rector, Bousfield's paper "On the homology spectral sequence of a cosimplicial space" constructs a spectral sequence which takes in a cosimplicial space (here ...
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Is there a n-category structure on algebras for $e_n$-like operads?
I'm fishing in troubled waters here and therefore the question is vague and meant to be as general as possible. In particular "$e_n$-like operad" can be an algebraic or topological $e_n$ operad, as ...
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Chains and homotopy type
Let $$C^{\ast}:\mathbf{sSet}\rightarrow E_{\infty}\text-\mathbf{dgAlg}$$
be the cochain contravariant functor from the category of simplicial sets to the category of $E_{\infty}$-dg-algebras (over $\...
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Geometric interpretation of nonconnective, non-coconnective chain complexes / spectra?
Let's stipulate that
Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about ...
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Algebras of the cone monad on Top?
Let us work in Top, the category of topological spaces - although the reader is welcome to replace this by their favorite convenient category of topological spaces.
If $X,Y$ are spaces, let $X\ast Y$ ...
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Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?
Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...
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The interaction between differentials on a graded ring and chain-homotopy equivalences
I am wondering about the following question:
Given a differential graded algebra $A$, how many other differentials can we put on the underlying graded ring of $A$, which are also chain-homotopy ...
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$\ell$-adic Eilenberg-MacLane space and Brown representability
I posted the following question on MathStackexchange, where it was suggested that I should move my question to Mathoverflow, which do here (https://math.stackexchange.com/questions/3550741/algebraic-...
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A fiber bundle of the Euclidean space over an orbifold
Consider a fiber bundle $p: F\hookrightarrow
E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...
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Homology of a semisimplicial scheme
This is a question about the homology of a complex made of algebraic varieties. Consider the following subgroups of $\mathrm{SL}_3$ (defined over $\mathbb{Z}$).
$$
P_{1,2} = \left\{\left(\begin{...
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Group homomorphisms between Eilenberg-MacLane spaces
It is well-known that for two (discrete) abelian groups $G$ and $H$, the set $[K(G,n),K(H,n)]_*$ of based homotopy classes of maps between the corresponding Eilenberg-MacLane spaces is in canonical ...
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Shapes of cores of symmetric monoidal $(\infty,n)$-categories (with duals)
According to the cobordism hypothesis, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-categories with duals, then framed fully extended TQFTs with target $\mathcal{C}$ are an $\infty$-groupoid, ...
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Compactly supported cohomology of a topological DM stack
Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...
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Homotopy type of a $4$-manifold with finite fundamental group (paper by S. Bauer)
I'm studying Stefan Bauer's paper
The homotopy type of a 4-manifold with finite fundamental group. In: tom Dieck T. (eds) Algebraic Topology and Transformation Groups. Lecture Notes in Mathematics,...
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160
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The fundamental loopoid?
Let $X$ be a homotopy type (modeled as either a topological space or a simplicial set). We can construct a category as follows: The objects are maps $f,g : S^1 \to X$. A morphism $f \to g$ is a map $S^...
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Continuous functors, spectra and homology theories
Let $T:\mathbf{Top}_*\to \mathbf{Top}_*$ be a continuous functor and $E$ a spectrum with maps $\sigma_n:E_n\wedge S^1\to E_{n+1}$. We have a new spectrum $TE$ with structure maps
$$(TE_n)\wedge S^1\...
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The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces
Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.
N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from
https://neil-strickland.staff.shef.ac.uk/courses/...
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Cardinalities associated to the Bousfield lattice
By Ohkawa's theorem, the Bousfield lattice $B$ (of the $\infty$-category of spectra) is a small, complete lattice with $2^{\aleph_0} \leq |B| \leq 2^{2^{\aleph_0}}$ (the exact cardinality is an open ...
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245
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Different algebra-structures on $\operatorname{THH}(\mathbb F_p)$?
By definition, we have a ring map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$. Post-compose with the canonical map $\mathbb Z_p\to\mathbb F_p$, we get a ring map $\mathbb Z_p\to\operatorname{THH}(\...
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Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)
$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
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Finite generation of the image of the induced homomorphism on homotopy groups of infinite loops spaces
Let $f:X\rightarrow Y$ be a map of infinite loop spaces such that image of homology groups $H_i(X,\mathbb{Z})$ for $i\geq 1$ under $f_*$ are finitely generated. Does this imply that the image of ...