Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas ...

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$K_0$ an $KH_0$ of a normal crossing variety

Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...
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1answer
179 views

Different model structures on Top

There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure. In the Mixed model structure ...
12
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1answer
432 views

Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line)

If $A$ is an abelian group, we have $Aut\left(K\left(A,n\right)\right)=Aut(A) \ltimes K\left(A,n\right),$ where the left hand side is the space of self-homotopy equivalences. Is there an easy way to ...
3
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0answers
98 views

Symmetric sub-simplicial sets of the nerve of $\mathbb{N}$

The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$ degeneracies: inserting $0$ faces: adding consecutive entries ...
12
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1answer
208 views

Computing Homotopy Fixed Point Spectral Sequences related to Morava E thories

Given a finite subgroup of $G$ sitting inside the Morava stabilizer group $S_n$, we can form the homotopy fixed point spectrum $E_n^{hG}$. There is a spectral sequence with $E_2^{s,t} = ...
14
votes
3answers
863 views

Motivation for equivariant homotopy theory?

I'm in the process of learning equivariant homotopy theory, so I was wondering: what is the importance of equivariant homotopy theory, and what has it been applied to so far? I know of HHR's solution ...
7
votes
0answers
377 views

Algebraic geometry introduction for homotopy theorists/algebraic topologists

Algebraic geometry has a plenty of decent introductory texts now. Some are of the classical commutative algebraic approach(following EGA), like Ravi Vakil's "Foundations". Some use facts from ...
3
votes
1answer
187 views

geometric conditions on maps between manifolds inducing monomorphisms on cohomology

Let $M,N$ be manifolds whose dimensions may be different. Let $f: M\longrightarrow N$ be a smooth map. What geometric conditions on $f$ can we impose such that the induced homomorphism $$ f^*: ...
4
votes
1answer
179 views

Parametrized Dold-Kan correspondence?

The stable Dold-Kan correspondence says that for every commutative ring $R$, there is an equivalence of $\infty$-categories between the category $Ch(R)$ of (unbounded) chain complexes of $R$-modules ...
3
votes
1answer
77 views

Homotopy classes of homeomorphisms of a multiple pointed space

Let $M$ be a multiple pointed space, i.e. $M$ is a topological space and there is a finite point set $M\supset P=\{p_1,...,p_k\}, k<\infty$. Such a $p_i$ is called a marked point. A map ...
7
votes
1answer
169 views

Fiber vs homotopy fiber in model categories: simple question

I have a concrete problem with the homotopy fiber and I am getting lost with the literature. I state my question and, to avoid confusions, I state downwards the definitions I am using. Let $C$ be a ...
7
votes
0answers
168 views

When do contributions of $\pi_*(L_{K(n)}S^0)$ to $\pi_*(S^0)$ stabilize?

I apologize if this question is obvious as I am just beginning to learn this field. I am also abusing the dependance on $p$ for notational ease. The Chromatic Convergence theorem establishes that ...
10
votes
2answers
372 views

In CGWH, is every cofibration an inclusion with closed image?

As the title suggests, in CGWH, is every cofibration an inclusion with closed image?
9
votes
4answers
419 views

Vietoris-Begle theorem for simplicial sets

I've learned the theorem when reading a comment by Vidit Nanda to my question see here. Here is the (simplified) version of the theorem for topological spaces: Vietoris-Begle Theorem Let ...
4
votes
1answer
166 views

Localization at the Johnson-Wilson spectrum and rationalization

Is there a clean proof that the $L_n$, localization at $E(n)$, is simply rationalization (i.e. $L_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this here, but I haven't seen an answer.
0
votes
1answer
188 views

fiber, homotopy fiber of spaces

Suppose we have a pullback of topological spaces (CW-complexes) $B\rightarrow A\leftarrow C$ which I will denote by $D$. Assumptions The induced map $D\rightarrow C$ is a trivial fibration The map ...
7
votes
1answer
266 views

What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?) By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
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0answers
223 views

Generalization for Leray Hirsch theorem for Principal $G$-bundle [closed]

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
13
votes
1answer
194 views

Free generators for the fat commutator subgroup

There is a homomorphism $\langle x,y\rangle\to\langle x\rangle$ of free groups, sending $y$ to $1$. We can combine this with the other obvious homomorphism to get a surjective homomorphism $$ ...
8
votes
1answer
232 views

Why does strong convergence of the EMSS imply that Tot commutes with suspension spectrum?

Given a fiber square of simplicial sets $$\begin{array}{cc} & \hspace{-7mm} E \\ &\hspace{-7mm}\downarrow \\ \ast\longrightarrow &\hspace{-7mm} B \end{array}$$ ...
3
votes
1answer
193 views

Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this. I wonder if there is any source telling about a ...
11
votes
2answers
594 views

Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?

Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories? That is, for each spectrum $E$, do we have a lift in the ...
7
votes
1answer
251 views

Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...
3
votes
1answer
134 views

very basic question on p-typical group law and MU and BP

I am a beginner trying to understand Dan Quillen's idempotent map $\xi:MU_{(p)} \to MU_{(p)}$ described in his classic paper 5-page paper On formal group laws... Let $F$ be a formal group law over a ...
6
votes
1answer
345 views

Iterated Homotopy Quotient

If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...
2
votes
0answers
95 views

Sufficient criteria for a nerve of a topological category to be good

I know that the following statement is true and I am looking for a reference: Given a topological category $\mathcal C$ (i.e. morphisms and objects form a space and all maps in the definition of a ...
9
votes
1answer
236 views

Are $E_n$-operads not formal in characteristic not equal to zero?

This is a short question: Is it just unproven folklore (yet), or is it definitively known that $E_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?
3
votes
2answers
272 views

Is it possible to compute coefficients of the formal group of an elliptic curve?

This may be trivial but I cannot find it. Given an elliptic curve, $C$ over $R$ with chosen parametrization $R\to \mathbb{A}^5_{\mathbb{Z}}=Spec(A)$, is there a way to compute coefficients of the ...
5
votes
0answers
144 views

On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...
18
votes
2answers
522 views

Why study the p-completions of a space?

Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...
6
votes
0answers
199 views

How do the direct and inverse image sheaf functors interact with homotopy?

This is a crosspost of this MSE question. The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by ...
13
votes
1answer
247 views

Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphisms of the first

In a previous Mathoverflow question, we saw that the fundamental group of the space $Imm(S^2,\mathbb{R}^3)$ of immersions the 2-sphere in ordinary 3-space is isomorphic to $\mathbb{Z}/2 \times ...
13
votes
1answer
350 views

Fibrant-cofibrant models of Eilenberg-MacLane spectra

There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...
3
votes
1answer
101 views

Geometric Realizations of Simplicial Based Spaces

Suppose I have two simplicial based topological spaces $X_\bullet$ and $Y_\bullet$, and the degeneracy maps of each satisfy the based homotopy extension property (but not necessarily the unbased ...
13
votes
2answers
769 views

Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological ...
13
votes
3answers
271 views

How stable is the top cell of a Lie group?

It is well known that the fundamental class of a compact Lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable ...
8
votes
1answer
209 views

Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.] I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...
4
votes
4answers
484 views

When is the quotient by an $n$-fold loop space an $m$-fold loop space?

Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about ...
27
votes
1answer
558 views

Classifiying sphere eversions

For a year I have been giving lectures on a (probalby) new way to present an explicit sphere eversion. These lectures include a review of many other explicit eversions that have been described, as ...
1
vote
1answer
91 views

dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32 Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...
18
votes
1answer
350 views

Del Pezzo surfaces and homotopy groups of spheres

A (complex) del Pezzo surface is a smooth projective complex surface with ample anticanonical line bundle. Such surface has a degree defined as the self intersection of the canonical divisor. It is ...
3
votes
1answer
128 views

cohomology ring of infinite iterated loop space

What is the cohomology ring $$ H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)? $$ I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...
3
votes
1answer
98 views

unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then $$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence $ (*) $ $$ ...
4
votes
1answer
183 views

Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...
26
votes
2answers
1k views

What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is $\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...
39
votes
1answer
1k views

Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected? Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
0
votes
1answer
246 views

p-local space vs p-completion.

I have some trouble to understand the difference between the p-completion and p-local space. if $X$ a simply connected spaces such that all higher homotopy groups are finitely generated groups then ...
5
votes
1answer
208 views

Representing classes in *relative* homology by submanifolds

There are nice results for representing homology classes by submanifolds, in particular for any class in $H_i(X)$ with $i\le 6$, see here. When $X$ is low-dimensional I can start getting explicit, but ...
2
votes
1answer
89 views

Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors $F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...
3
votes
1answer
156 views

Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...