# Tagged Questions

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 ...

46 views

I am reading the book: Convex Integration Theory by D. Spring and encounter a question which I subtract as follows. Let $p: X \rightarrow V$ be a smooth fibre bundle and $\mathcal{R} \subset ... 2answers 713 views ### The homotopy category is not complete nor cocomplete I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts. What ... 0answers 69 views ### Computing the order of elements in a non abelian exterior square of a finite group If we have an explicit group$G$, and we pick two elements$g,h \in G$, could we find the order of the element$g \wedge h \in G \wedge G$? The best thing I could find is Theorem 1.1 in Ellis' Book ... 3answers 488 views ### Maps with Hopf invariant zero are suspensions Let$h:\pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if$x \in \pi_{2n-1}(S^n)$satisfies ... 0answers 85 views ### Universal enveloping algebra functor preserves quasi-isomorphism Let$k$be a field of characteristic 0. Let$\mathtt{DGA}_{k}$denote the category of DG algebras and$\mathtt{DGLA}_{k}$denote the category of DG Lie algebras. It is well known that there are model ... 1answer 322 views ### Are two equivariant maps between aspherical topological spaces homotopic? Let$f: X \rightarrow Y$be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then$\pi_1(X)$acts on the universal cover of$X$via deck transformations, and on ... 2answers 329 views ### Homotopy of space of immersions, Smale-Hirsch theorem If$M$and$N$are manifolds with$\dim M< \dim N$, we denote by$Imm\left(M,N\right)$the space of immersions of$M$in$N$. Let$M$and$M'$manifolds of dimensions$m>0$. It is true that if ... 2answers 460 views ### Reference request: Goodwillie tower of the identity The Taylor (Goodwillie) tower of the identity functor on based spaces has as its$j$-th layer the infinite loop space-valued functor $$X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]})$$ in ... 2answers 330 views ### Simplest explicit counterexample for$Vect(BG) \ne Rep(G)$as monoids Let$G$be a topological group,$Vect(BG)$the monoid of complex vector bundles over its classifying space (not the stack!) and$Rep(G)$its monoid of complex representations. Generally$Vect(BG) \ne ...
190 views

I was looking for particular and explicit examples of $E_\infty$-operads. I know the $E_\infty$-operad defined by Smith in http://arxiv.org/abs/math/0004003, and the Barratt-Eccles operad, but it is ...
91 views

### Cofibrations in the model structures for non-negative graded (commutative) DG algebras

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}_{k}^{+}$ denote the category of non-negative graded ...
152 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...
94 views

### Smoothing a continuous section in 1-jet bundle

Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle). I ...
143 views

### Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map ...
157 views

### Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...
630 views

### Étale homotopy type of non-archimedean analytic spaces

The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway. I am wondering about basic facts of what would naturally ...
41 views

### Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property. Here is the definition of f.p.s.p.("map" means ...
349 views

### Table of (integral) cohomology groups of K(Z,n)

Can I find somewhere a table of the (first few) cohomology groups of $K(\mathbb{Z},n)$ with integer coefficients? It seems like a natural counterpart to the table of the homotopy groups of spheres, ...
668 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 ...
869 views

### What is the relation between sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab). Are there some ...
183 views

### Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...
Given a category $\mathcal{C}$ with a notion of covering $\{ U_{i} \rightarrow X \}$ for an object $X$ (say $\mathcal{C}$ is a Grothendieck site), we can form the Cech nerve $$\cdots ... 0answers 116 views ### Is every path connected space continuously path connected Recall a topological space X is path connected if for all x,y \in X there is a continuous function f\colon [0,1] \to X such that f(0)=x and f(1)=y. Say that X is continuously path ... 1answer 247 views ### Difficulties with descent data as homotopy limit of image of Čech nerve Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ... 2answers 414 views ### Discrete Morse theory: how do zig-zag paths determine homotopy type? Let \Delta be a simplicial complex (or more generally, a regular CW complex). Let \mathcal{M} be a Morse matching (or equivalently, a discrete Morse function) on \Delta. By Forman's theorems, ... 1answer 88 views ### Can a weak fibration category be non saturated? A weak fibration category is a category \mathcal{C} equipped with two subcategories$$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$containing all the isomorphisms, such that the following ... 0answers 152 views ### Complete Segal operads and dendroidal sets There is a Quillen equivalence between the model category presenting Lurie's \infty-operads (which are inner fibrations \mathcal{C}\to\mathrm{N}(\mathbf{F}) satisfying certain conditions) and the ... 1answer 150 views ### Formula relating the cup product in dimensions n and n+1 Let's will write K_n for the Eilenberg-MacLane space K(\mathbb{Z},n). I remind that K_n is equivalent to the loop space of K_{n+1}. Let’s consider the map \smallsmile:K_n\times K_m \to ... 1answer 201 views ### (Geometric) Proof for the projective bundle formula in K-theory I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove: Projective bundle formula: Let \pi: E \to X be a ... 1answer 317 views ### Why are quasi-isomorphisms of homotopy algebras only defined for arity 1? When reading about homotopy algebras (e.g. L_\infty-algebras, A_\infty-algebras), an \infty-morphism f is called an \infty-quasi-isomorphism if f_1 is a quasi-isomorphism. Recall/Example ... 2answers 372 views ### H-space structures on non-sphere suspensions? It is well known that S^n admits an H-space structure if and only if n=0,1,3,7. I'm interested in whether there are other suspensions \Sigma X that admit H-space structures: Question 1 For ... 2answers 2k views ### Generalized Categories for “Higher Homotopy Groupoids” I was thinking about the definition of higher homotopy groups \pi_n of a topological space in comparison to the common extremely formal fundamental groupoid construction of \pi_1. I'd like to be ... 3answers 1k views ### How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory? The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the n-sphere, and Adams proved that this is actually always the best ... 1answer 236 views ### Closed formulas for topological K-theory? Let X be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-K-theory. Let E be a complex vector bundle with a given line ... 0answers 169 views ### Homotopy groups of the Grassmannians What are the homotopy groups of the oriented Grassmannian Gr^{+}(p,q) (p-planes in R^{p+q}) \pi_{r}(Gr^{+}(p,q)), r \le pq? Do you know any references on the web about it? 1answer 230 views ### From relative categories to marked simplicial sets Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two ... 0answers 141 views ### The homotopy type of the mapping space Map_{B\rho}(BS^1,BG)? for G a compact Lie group Given a homomorphism \rho:S^1\rightarrow G with G a compact Lie group there is an induced map of classifying spaces B\rho:BS^1\rightarrow BG. What is known about the homotopy type of the mapping ... 1answer 323 views ### Homotopy fiber of a map between classifying spaces I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong): Let G and H be topological groups and f : G \to H be ... 2answers 378 views ### When do colimits agree with homotopy colimits? I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim(F)=colim(F_c) where F_c is the cofibrant replacement of F. However, it ... 2answers 417 views ### RO(G)-graded homotopy groups vs. Mackey functors Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work \infty-categorically. Also, I've looked through other similar MO questions, but I didn't find ... 6answers 1k views ### A conceptual proof that local fibrations over paracompact spaces are global fibrations? I'm teaching some homotopy theory at the moment, and I'm discovering a number of things I've never before learned properly. (I guess everybody who's ever taught knows that feeling.) One of those ... 1answer 230 views ### Relative version of Quillen's theorem A Quillen's Theorem A is formulated as follows: Let F:X\to Y be a functor between small categories. Suppose for each y\in Y the category F/y is contractible. Then F induces a weak equivalence ... 3answers 5k views ### What is the “intuition” behind “brave new algebra”? Y.I. Manin mentions in a recent interview the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ... 2answers 2k views ### What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ... 8answers 1k views ### How to get product on cohomology using the K(G, n)? This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map ... 0answers 65 views ### SImple homotopy type of a mapping cone Consider two CW complexes A and B. Let f be a continuous map from A to B. Take a cone of f and denote it by Cone(f). If homology complex of Cone(f) is acyclic, one can identify homologies of A and B. ... 0answers 394 views ### Solving polynomial systems with homotopy. Where is the bottleneck? I have a polynomial system with n+k unknowns (n+k can be greater than 8), that is known to have a limited number of isolated solutions. I want to solve this system numerically, but if I plug it ... 1answer 2k views ### Are there any “homotopical spaces”? This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense. [Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ... 1answer 252 views ### Whitehead products and Framed Manifolds The attaching map for the top cell of the torus S^n \times S^n is a map$$ [x,y]: S^{2n-1} \to S^n \vee S^n  where the notation is such that $x,y : S^n \to S^n \vee S^n$ are the two ...
Suppose that $X$, $Y$ and $Z$ are topological spaces, with $A\subset X$, a map $f:A\rightarrow Y$, and a homotopy equivalence $\phi:Y\rightarrow Z$. It seems fair to think that the adjunction spaces ...