Questions tagged [at.algebraic-topology]

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

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4
votes
2answers
246 views

Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid

Let $X$ be a path-connected smooth manifold, it is known that: $$H^1(X):=H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R).$$ Explicitly, a closed one-form $\alpha$ gives a function on $\pi_1(X)$ by $[\...
7
votes
1answer
412 views

Categorical Significance of Fibrations

It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
8
votes
3answers
873 views

Link of a singularity

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$. If we set $x = x_1+ix_2, y = y_1+iy_2, z ...
4
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0answers
174 views

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/...
1
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0answers
111 views

Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
8
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0answers
209 views

Are the braid groups good in the sense of Toën?

In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
6
votes
1answer
263 views

Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
7
votes
1answer
343 views

Obstruction to homotopy, cohomology operations and Dold-Whitney theorem

I am reading the famous paper by Dold and Whitney "Classification of Oriented Sphere Bundles Over A 4-Complex". I'll state their theorem for the case of SO(3) bundles Classification Theorem:Let $B_1,...
10
votes
1answer
259 views

Functoriality of infinite loop space machines?

If $C$ is a symmetric monoidal category, then $BC$ is canonically an algebra over a certain $E_\infty$ operad, but if $F: C \to D$ is a symmetric monoidal functor then (as far as I can see) $BF: BC \...
3
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0answers
152 views

Reference Request: Categorical/Functorial approach to Formal Schemes and Formal Groups

For my Bachelor thesis, I am looking for refernces about Formal Schemes and Formal Groups from the categorical point of view. So far I was only able to find Strickland's article Formal Schemes and ...
2
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0answers
68 views

Discrete Morse theory, choice of Morse function, and removing noise

If I have a simplicial complex, and a discrete Morse function defined on the simplices, I can use persistent homology to produce a barcode which helps me distinguish "persistent" shape from noise. To ...
5
votes
0answers
94 views

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 ...
8
votes
3answers
245 views

Homotopy type of non-Cohen-Macaulay complexes

Most interestingly defined (pure) simplicial complexes that occur in topological combinatorics are Cohen-Macualay. Some of these are even shellable (or have the homotopy type of a wedge of spheres; of ...
5
votes
1answer
185 views

Curvature and asphericity of cube complexes

Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...
4
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0answers
104 views

Mysterious identity in cosimplicial $R$-module with Lie brackets

I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
8
votes
2answers
469 views

Other homotopy invariants?

The idea of using maps from a sequence of simple standard objects into a topological space $X$ as $probes$ to explore its topology is ubiquitous. One organizes these maps into equivalence classes in ...
7
votes
1answer
746 views

Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds. Cobordism Theory for DIFF/Differentiable/smooth manifolds However, there are ...
9
votes
1answer
385 views

What do absolute neighborhood retracts look like?

In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...
8
votes
0answers
111 views

Non-additivity of intersection forms

Given two oriented $4k$-manifolds $X_1$ and $X_2$, Novikov additivity tells us that $$ \sigma(X_1 \sharp X_2) = \sigma(X_1) + \sigma(X_2).$$ More generally, if we glue the boundaries of two such ...
4
votes
0answers
88 views

Homotopy colimits of long sequences

Let $\lambda$ be a limit ordinal, and let $F: \lambda\to \mathcal{T}_*$ be a diagram of pointed spaces with shape $\lambda$. Write $X = F(0)$ and $Y = \mathrm{hocolim} F$. I believe it to be true (I ...
6
votes
1answer
346 views

A finite p-group question: can this happen?

Let all groups here be finite $p$--groups. Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 \lhd L_1 \lhd \cdots \lhd L_r = H$, such that each $L_i/...
16
votes
1answer
877 views

A finite 2-group containing the dihedral group of order 16?

The dihedral group $D_{16}$ of order 16 has a presentation $D_{16}= \langle a,t \ | \ a^2=t^8=atat=e\rangle$. Question: Does there exist a finite 2-group $G$ containing $D_{16}$ as a subgroup, and ...
15
votes
1answer
489 views

Characteristic classes of symmetric group $S_4$

For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
8
votes
0answers
253 views

Lusternik-Schnirelmann Category of 4-Manifolds

Whilst reading 'the book' I stumbled across the following elegant little theorem, due to Gómez-Larrañaga and González-Acuña. Let $M$ be a closed $3$-dimensional manifold. Then its Lusternik-...
5
votes
1answer
256 views

Using Stiefel-Whitney class to build new principal bundles

I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me. Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ ...
6
votes
1answer
226 views

What are orbifolds with corners?

What is the geometric definition of orbifolds with corners? Here “geometric" means that there is a definition in chapter 8 of the draft of Dominic Joyce's book D-manifolds and d-orbifolds: a theory of ...
6
votes
2answers
402 views

Dold-Kan correspondence in the category of symmetric spectra

The Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes of abelian groups is a classical result. It states that the ...
4
votes
1answer
307 views

Definition of geometric monodromy

Consider a polynomial $f \in \mathbb C[x_1,\dots ,x_n]$. An atypical value of $f$ is a complex number about which $f:\mathbb C^n\to \mathbb C$ is not a topological fiber bundle. Writing $\mathrm{Atyp}(...
3
votes
1answer
199 views

A model for the framed little disks operad $f{\cal D}_n$ with arity one *equal* to $SO(n)$?

The framed little disks operad $f{\cal D}_n$ can be described as the semidirect product ${\cal D}_n \rtimes SO(n)$, where ${\cal D}_n$ is the little disks operad and $SO(n)$ is the special orthogonal ...
4
votes
1answer
197 views

Homotopy type of G-CW-structure

Let $G$ be a finite group, and $X$ be a free $G$-space. Moreover, assume that $X$ has a homotopy type of a CW-complex. Does $X$ have $G$-homotopy type of a $G$-CW complex also? Edit: My main ...
8
votes
1answer
229 views

When does $BG \to BA$ loop to a homomorphism?

If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
3
votes
1answer
212 views

Topology of functional spaces

Let $X$ be a finite CW-complex of dimension $n$. Fix an natural number $k < n$, and let $M(X, \mathbb{S}^k)$ be the space of all continuous function from $X$ to the k-sphere $\mathbb{S}^k$ endowed ...
5
votes
0answers
193 views

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}(\...
4
votes
2answers
412 views

Background needed to understand modern research on knot homology theories

I am a student of mathematics, and have some background in Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff), Differential Geometry (Lee, Kobayashi-Nomizu), Riemannian Geometry (Do Carmo), ...
3
votes
1answer
178 views

Wall self-intersection invariant for odd-dimensional manifolds?

I am trying to convince myself that a naïve definition of the Wall self intersection number should not work for odd-dimensional manifolds. Namely, let $X^{2n-1}$ be a smooth oriented closed manifold ...
3
votes
0answers
105 views

Naturality of Poincaré–Lefschetz

Let $X$ be compact and Hausdorff, $A\subseteq B\subseteq X$ both closed such that $X\setminus A$ is an open orientable $d$-manifold. Then also $X\setminus B$ is an open orientable $d$-manifold. We ...
3
votes
0answers
89 views

Extensive survey of computations of equivariant stable stems

Where can I find a comprehensive survey of computations of equivariant stems? To my knowledge, the status is: Classical Work of Araki and Iriye, Osaka J. Math. 19 (1982). ...
2
votes
2answers
267 views

Turning injection of homotopy groups to an isomorphism

Assume we have a connected CW-complex $Y$ and $X\hookrightarrow Y$ a connected sub-complex. We know that the inclusion induces injection on all homotopy groups. Is it true (or under what conditions it ...
7
votes
0answers
251 views

Reference request: complex K-theory as a commutative ring spectrum

Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum? For real $K$-theory I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (...
6
votes
2answers
466 views

Lecture notes by Mahowald and Unell

I'm trying to find lecture notes of Mahowald and Unell, titled "Lectures on Bott periodicity in stable and unstable homotopy at the prime 2". Does anyone happen to know if a copy exists online (and if ...
6
votes
4answers
638 views

On fundamental groupoid of fundamental groupoid

Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi_1(X)$. Here, the fundamental groupoid $\Pi_1(X)$ is made into a topological groupoid giving a topology on the ...
6
votes
1answer
165 views

Visualizing a Whitehead product: the attaching map $S^3\to S^2\vee S^2$

There are informative and easily accessible images and videos that illustrate the Hopf fibration $S^3\to S^2$ by describing what happens to the fibers in the unit cube $(0,1)^3\approx S^3\backslash \...
2
votes
0answers
253 views

Novikov conjecture

The statement of the Novikov conjecture is a bit esoteric. Does the following simplified conjecture have any known counterexamples? C: For a smooth closed 4n-fold $M$, the Pontryagin numbers are ...
4
votes
0answers
76 views

Lower Semicontinuity of the Fundamental Group

Gromov-Hausdorff convergence of compact metric spaces certainly do not preserve topology. However, at the level of fundamental group, for length spaces the following result is well known (I think): ...
8
votes
1answer
198 views

Splitting low-dimensional $p$-local CW complexes for large $p$

Fix a prime $p$. I have a sketch of a proof that if $X$ is a finite simply-connected CW complex with $\mathrm{dim}(X) < p$ then for some $t\in \mathbb{N}$, the $p$-localization $\Sigma^t X_{(p)}$ ...
8
votes
0answers
341 views

Homology of a quotient space defined by an equivalence relation

Let $(X,x)$ be a pointed connected CW-complex. Let $f:X\rightarrow X$ be a (pointed) homeomorphism. Denote $Y=X\vee X$ and $Y^{ n}=Y\times\cdots \times Y$ $n$-times. Lets define a new homeomorphism $h:...
3
votes
0answers
89 views

Methods for constructing or checking for nontrivial classes in de Rham cohomology with local coefficients

Let $M$ be a smooth manifold (possibly with boundary), $E \to M$ a flat vector bundle, and $\mathcal{L}$ the corresponding sheaf of parallel sections. Given a de Rham cohomology class $[\omega] \in H^...
9
votes
1answer
242 views

What is the homotopy fiber of $X \to X_{hG}$, where this is a pointed homotopy orbit?

The unpointed version is easy: the model $X = EG \times X \to (EG \times X)/G = X^{un}_{hG}$ is a fibration with fiber $G$. But when we go pointed, $X = EG_+ \wedge X \to (EG_+ \wedge X) / G = X_{hG}$ ...
21
votes
1answer
405 views

When does rationalization commute with homotopy fixed points?

Let $X$ be a $G$-space. There are a number of places in the literature where one can find the claim that under certain conditions rationalization and taking homotopy fixed points with respect to a ...
3
votes
1answer
161 views

$\pi_1$ action on relative homotopy groups $\pi_n(X,A)$

It is known that there is a natural $\pi_1(X,x)$ action on $\pi_n(X,x)$ which also induces a bijection $\pi_n(X,x)/\pi_1(X,x) \cong [S^n, X]$. Now, let $(X,A)$ be a pair of path-connected spaces and $...