19

This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a homology theory from a spectrum $X$ is to consider $\pi_*(X\otimes -)$, note that for compact $X$ your $[X,-]$ is of this form as well, since $$ [X,-] = \pi_*(DX\...


18

I want to mention five directions where in the last years significant progress has been made in chromatic homotopy theory. This is of course not exclusive! Unstable chromatic homotopy theory Among the greatest functors in existence is the Bousfield-Kuhn functor $\Phi_n\colon \mathcal{S} \to \mathrm{Sp}$ from spaces to spectra. The composite $\Phi_n\Omega^{\...


11

Tyler's comment answers my question. A bit more detail: the Postnikov tower of $k(n)$ is an Adams resolution, because the `bottom class' map $k(n) \rightarrow H\mathbb F_p$ is onto in mod $p$ cohomology; indeed $H^*(k(n);\mathbb F_p) = A_p//E(Q_n)$. The appendix by Greenlees and May has the details that two spectral sequences converging to $\pi_*(Y \...


8

A 1968 paper in Topology by Anderson and Hodgkin shows that $KO^*(K(\mathbb F_2, n)) = 0$ if $n \geq 2$. This implies that if $n \geq 2$, then no nonzero classes in $H^*(K(\mathbb F_2,n);\mathbb F_2)$ are SW classes. (And of course, $BO(1) = K(\mathbb F_2,1)$.)


8

Robert Thomason was the first person to draw attention to this question, before derived schemes and infinity categories. I believe that he proved that for a quasi-compact and quasi-separated scheme that $D_{qc}=\textrm{Ind}(\textrm{Perf})$. For example, see Thomason-Trobaugh section 2.3, though at first glance it appears that only proves the weaker statement ...


7

Any fibrant replacement for $S^n$, $n \geq 1$ is going to have infinitely many non-degenerate simplices. This is simply because there are infinitely many elements of $\pi_nS^n$. So, even though to a mathematician it seems that we can "compute" a fibrant replacement, it is not actually easy to program it in such a way that we can determine the homotopy ...


6

The mod $2$ cohomology of $S^0/2 \wedge S^0/2$ is a $\mathbf{F}_2$-vector space on generators in degrees 0, 1, 1, and 2. The classes in degrees 0 and 2 are connected by a nontrivial $\mathrm{Sq}^2$, so you cannot split $S^0/2$ off (any shift of) $S^0/2 \wedge S^0/2$. The topological version of this statement is the fact that there is a cofiber sequence $$S^1 ...


6

(This answer is written in a model-independent fashion -- translate to your favourite formalism). For every path $\gamma:[0,1]\to B$ you get an isomorphism in the homotopy category $X_{\gamma0}\xrightarrow{\sim} X_{\gamma1}$ (where with $X_b$ I denote the homotopy fiber over $b\in B$). Probably the easiest and most geometric way of constructing it is to ...


6

Pramod Achar is working on a book on perverse sheaves and applications in representation theory. It's a great book!


6

I'm currently taking a course on perverse sheaves and we are using Kashiwara & Schapira's Sheaves on Manifolds (published by Springer). It has all the things you mention and I've found it very readable!


6

Here are two possibilities: Topological Invariants of Stratified Spaces by Markus Banagl Intersection Homology & Perverse Sheaves: with Applications to Singularities by Laurenţiu G. Maxim


6

Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement. Computing the fibrant replacement for simplicial sets is quite easy: it is given by the Kan fibrant replacement functor Ex^∞. Explicitly, n-simplices in the fibrant replacement of a simplicial set X are ...


5

Let $A$ be an $\mathbf{E}_1$-ring, and let $x\in \pi_n A$. There are two distinct cases to consider. First, if $n = 0$, then the answer to your question is that $x$ is in the image of an $\mathbf{E}_1$-map from an $\mathbf{E}_\infty$-ring. Indeed, then $x:S^0\to A$ extends to a map $\Sigma^\infty_+ \mathbf{Z}_{\geq 0}\to A$, essentially because $B\mathbf{Z}_{...


4

I found it on professor May's web site at http://math.uchicago.edu/~may/PAPERS/42.pdf Since this link might disappear, it is also archived at the Wayback Machine.


4

Let $f:E\to B$ be a map of based spaces, and let $F$ be the homotopy fiber. Here is another way of constructing the action of $\Omega B$ on $F$. By definition, there is a homotopy pullback square $$\require{AMScd} \begin{CD} F @>>> \ast\\ @VVV @VVV \\ E @>>> B.\\ \end{CD}$$ Taking homotopy pullbacks along the inclusion $\ast\to B$ produces ...


3

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you ...


3

Given $C$ a small category (eventually, a small simplicial category) I denote by $UC$ the projective model structure on the category of simplicial presheaves on $C$ as in the paper. Using the kind of argument you have in mind we obtain the following theorem: Theorem: If $M$ is a simplicial model category, then there is an equivalence of categories between: ...


3

Assuming that $X$ is a smooth manifold, your question can be reformulated as: Under which conditions every self-homotopy-equivalence $X\to X$ is homotopic to a diffeomorphism? I will say that $X$ satisfying this property is smoothly rigid. I will say that $X$ is rigid if every self-homotopy equivalence is homotopic to a homeomorphism. Here are some ...


3

I believe these questions were studied and answered in the this 1984 paper: HIROSHIMA MATH. J. 14 (1984), 359-369 On the set of free homotopy classes and Brown's construction Takao MATUMOTO, Norihiko MINAMI and Masahiro SUGAWARA They also have counterexamples that seem to be the same as in the recent preprint of Arlin and Christensen that you mention. ...


3

The inclusion of groupoids into simplicial sets is fully faithful. Its left adjoint, $\Pi_1$ is given by left Kan extension of the functor $\Delta\to \mathcal{Gpd}$ sending the n-simplex to the contractible groupoid with objects $\{0,...,n\}$. The entirety of the data of the homotopy type of the space $X$ is contained in its singular simplicial set, which ...


2

If $p_0$ is not a critical point of $f$ then the implicit function theorem states that, there exists local coordinates $(x^1,\dotsc, x^n)$, defined in an open neighborhood $U$ of $p_0$ in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$ such that, in these coordinates we have ($m=\dim X$) $$ x^i(p_0)=0,\;\;\forall i, $$ $$ X=\{ x^{n-m+1}=\cdots =x^n=0\}, $$ $$ f(...


2

I have now (May 13) partitioned the answer into the blocks 1,2, as I think 2 is the simpler answer! 1 I hope the book Nonabelian Algebraic Topology will answer the question for you. A groupoid is level one of a structure called a crossed complex which is a kind of nonabelian chain complex but also with the groupoid structure in dimensions $\leqslant 1$...


1

This responds only to part of your post, but I think it's relevant to the basic premise. It is certainly not the case that all 2-knots have aspherical complements. In fact, the paper of E. Dyer and A. T. Vasquez, The sphericity of higher dimensional knots, Can. J. Math. 25 (1973), 1132–1136 shows that if a higher dimensional knots has aspherical complement ...


1

See Problem 4.23 and Problem 4.24 (with proofs) of Ulrich Bunke's Differential cohomology. The underlying abstract machinery for computing homotopy (co)limits via homotopy (co)ends is presented by Sergey Arkhipov and Sebastian Ørsted in Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories.


1

A very rough argument that can be (easily) formalized is as follows: We have a notion of $\infty$-groupoids. These are like groupoids, but they have homotopies between morphisms, homotopies between homotopies, and so on. Every topological space presents an infinity groupoid by taking the objects to be points, morphisms to be paths, morphisms between ...


1

I don't see how you could exclude that the gradient lies in the tangent space near the equator. Seems to me that you need to use the diffeomorphism provided by the Morse lemma applied to $f$ to straighten the level set of $f$ into a hyperplane locally. After that, work in the "straightened domain" and apply the Morse lemma to the distance to (the image of) ...


1

(The following was intended as a comment to Karol's answer, that after using the "Brown Type Factorization" trick, we can prove the result by applying a result in the book, which does not assume cofibrantness, but due to the word count, I think it might be more appropriate to present it as an answer.) First, using the Brown Type Factorization trick, we can ...


1

This may not be an answer but too long for a comment! Though I did not read your mentioned paper https://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf in details but what I understood from the Peter May's answer here Homotopy of functors that if you define a homotopy between 2 covariant functors $F,G :C \...


1

This has a partial positive answer. The theory of cyclotomic spectra using orthogonal spectra is developed in 'The homotopy theory of cyclotomic spectra' by Blumberg and Mandell. See https://arxiv.org/abs/1303.1694. Moreover, Nikolaus and Scholze have an exposition of the Böckstedt construction of THH using orthogonal spectra in their paper 'On topological ...


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