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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7
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1answer
91 views

Stable homotopy groups of $RP^{\infty}$

Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.
27
votes
5answers
2k views

What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
0
votes
0answers
61 views

Reference for generalized model categories [closed]

I would like to know some references for "Stanculescu's notion" of Generalized model category. Motivation The idea is find some works that use the notion and increase my knowledge about it. ...
6
votes
4answers
620 views

A reference for Calculus of Functors for Model Categories

I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories. I am at least aware of some of the extended ...
-4
votes
1answer
143 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
3
votes
2answers
331 views

Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...
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 ...
6
votes
0answers
78 views

Samelson Products in $SO(n)$

Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most ...
6
votes
3answers
998 views

Crossed module structure on homotopy groups.

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy $\partial(g\cdot c)=g(\partial c)g^{-1}$, and $cc'c=(\partial c)\cdot ...
16
votes
2answers
958 views

Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
6
votes
0answers
92 views

Detection tools for (reduced) suspension

I'm learning about loop spaces and the work of Stasheff on $A_{\infty}$-spaces. The broad idea that I'm getting is the following. Given a space $Y$, we want to know under which conditions there exists ...
4
votes
0answers
100 views

What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...
11
votes
3answers
1k views

Grothendieck's Homotopy Hypothesis - Applications and Generalizations

Grothendieck's homotopy hypothesis, is, as the $n$lab states: Theorem: There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$. What are the ...
6
votes
0answers
199 views

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 ...
8
votes
1answer
436 views

What is the Status of Borel conjecture today?

Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.
9
votes
0answers
262 views

What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?

Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...
4
votes
0answers
127 views

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to ...
25
votes
15answers
7k views

“Homotopy-first” courses in algebraic topology

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
6
votes
2answers
444 views

quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ ...
16
votes
1answer
535 views

(really) basic intuition for $\mathbb A^1$-homotopy theory

Apologies in advance if this question is inappropriate for MO. I'm trying to read here and there about $\mathbb A^1$-homotopy theory in algebraic geometry. I understand some abstract machinery is ...
2
votes
1answer
198 views

isotopy equivalence (topological meaning) between $CW$-complexes

Let $M$ and $N$ be $CW$-complexes. Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map $$ F: M\times ...
4
votes
2answers
239 views

homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary ...
6
votes
0answers
159 views

Homotopy classes of maps and cohomology classes (Hatcher, AT, Thm 4.57) [duplicate]

Hatcher's AT Theorem 4.57 is used in both the algebraic topology construction of Seifert surfaces, and the (similarly flavored) proof that given a compact 3-manifold (with or without boundary), we can ...
2
votes
1answer
69 views

Homotopy limits of homotopically constant diagrams over contractible categories

I suspect that the following result should be true and more or less well known: Let $\mathcal{M}$ be a model category and $I$ a small category with contractible nerve. For every diagram $X: I \to ...
10
votes
2answers
354 views

How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \rightrightarrows BG$ is $G/G$ (adjoint action)?

Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids with one vertex, labeled $1/G$, the one arrow from that vertex ...
6
votes
2answers
2k views

the hopf invariant of the hopf construction

I'm having some trouble with a problem about the Hopf construction, in the exercises for Ch. 4 of Mosher & Tangora. Given a map $g : S^{n-1} \times S^{n-1} \rightarrow S^{n-1}$, we get a map ...
15
votes
5answers
2k views

Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

Many topologists express a clear preference for working with CW complexes instead of simplicial sets. One of the reasons is that the cellular chain complex of a CW complex is often easier to work ...
29
votes
5answers
3k views

Analogue to covering space for higher homotopy groups?

The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched ...
4
votes
1answer
233 views

Homotopy classification of selfmaps of product of spheres?

Self-maps of n-torus $T^n=S^1\times ...\times S^1$ are classified by the induced homorphism of fundamental group $\pi_1 T^n=Z^n$. Is a similar result true form self-maps of $S^k\times ...\times S^k$ ...
20
votes
1answer
739 views

Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I

In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical ...
4
votes
1answer
127 views

Directed homotopy in the Cayley graph of a monoid

There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some ...
4
votes
0answers
68 views

Invert quasi-isomorphisms of symmetric cooperads

The theory of symmetric operads in chain complexes (say over a good enough field) is in some sense nice, because we have a well defined homotopy theory. In particular we have a notion of ...
1
vote
1answer
145 views

torsion part of the cohomology module of configuration spaces of manifolds

Let $M$ be a manifold and $C_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational ...
7
votes
0answers
162 views

A definition of the homotopy colimit of a coherent diagram

Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...
20
votes
2answers
2k views

Does this approach for the Poincaré conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. ...
3
votes
2answers
463 views

Principal bundles that can't be detected by spheres

The question I'm trying to answer is the following: Let $P \to X$ be a principal $G$-bundle (over a connected CW complex) satisfying that all pullbacks to spheres (of arbitrary dimension) are ...
10
votes
3answers
2k views

The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres

Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of ...
2
votes
0answers
167 views

Algebraic K theory, Karoubi completion and splitting

Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in ...
3
votes
0answers
348 views

Homotopy category of groupoids

The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description: iso-classes of functors. formally invert equivalence functors (i.e. ...
2
votes
1answer
129 views

Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...
7
votes
1answer
193 views

Is it known whether this space is a suspension space?

For each prime $p\geq 3$ let $\alpha_p:S^{2p}\to S^3$ denote a representative of $\pi_{2p}S^3$ of order $p$. Berstein and Hilton showed that for each $p$ the homotopy cofiber $C_{\alpha_p}$ of ...
9
votes
0answers
279 views

When does algebraic K theory behave like a cohomology theory

Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of ...
10
votes
2answers
625 views

Why are Delta-generated spaces locally presentable?

Does anybody understand why Delta-generated spaces are locally presentable? This is of course claimed by Jeff Smith, and there is a paper by Fajstrup and Rosicky A convenient category for directed ...
17
votes
7answers
3k views

Are there two non-homotopy equivalent spaces with equal homotopy groups?

Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth ...
13
votes
1answer
520 views

Motivation and potential applications of spectral algebraic geometry

Nowadays there is a lot of talk about derived algebraic geometry, but not so much about the related subject of spectral algebraic geometry. Now I'm curious what future is there for spectral algebraic ...
5
votes
0answers
243 views

What is an Elementary “Homotopy, Model” Topos?

Context: Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C. Thm ...
3
votes
0answers
117 views

Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
35
votes
6answers
2k views

Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?

In his 1967 paper A convenient category of topological spaces, Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces as a good replacement of the category Top topological ...
10
votes
1answer
431 views

teaching higher algebra

Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)? I'm asking out of curiosity (and also hoping for more resources). The ...
6
votes
4answers
506 views

What does “higher monodromy” tell us about a principal bundle

Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to ...