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\...


19

The course has been published in the Journal de Mathématiques Pures et Appliquées, volume 24 (1945), and can be found here: first part and second part and third part. A quote from Leray's obituary: The prisoners in the camp were mostly educated men, career or reserve officers, many of them still students. As in several other camps, a "university" was ...


8

Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles. Edit: The argument for this is as follows. Let $\mu : BTOP \times BTOP \to BTOP$ be the map corresponding to Whitney sum of (stable, topological) bundles. The question ...


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, $BO(1) = K(\mathbb F_2,1)$.)


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

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 ...


4

There are a number of easily stated problems in elementary computational geometry that have been solved only relatively recently, e.g., the origami existence theorem that a single rectangular sheet of paper can be folded into the shape of any connected polygonal region, even if it has holes; the fold-and-cut theorem that any shape with straight sides can be ...


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

A recent preprint of Boyde improves this bound, showing that $$\log_p(\#\pi_{n+k}(S^n)_{(p)}) \leq c_p 2^{k/(p-1)}$$ where $c_p = \frac{1}{4}2^{1/(p-1)}$ Note that this bound depends on $k$ and not $n$, so it stabilizes to show that $$\log_p(\#\pi^s_{k}(\mathbb S)_{(p)}) \leq c_p 2^{k/(p-1)}$$ In his introduction, Boyde mentions some earlier results of ...


3

(ii) I would like to prove that there are no complex surfaces that satisfy (ii). Indeed, suppose that the universal cover $\widetilde X$ of a complex surface $X$ is diffemorphic to $\mathbb C^2\setminus 0$. Let's prove that $\widetilde X$ is biholomorphic to $\mathbb C^2\setminus 0$. First, we note that $X$ has a finite cover that is diffeomorphic to $S^3\...


2

1) The set of continuous everywhere but differentiable nowhere functions on the unit interval is a meagre set of measure 1. 2) The existence of a space filling curve, or more generally surjective continuous maps $S^m \to S^n$ for $n>m$ (and then the fact that however any such map is homotopic to a map that misses a point).


2

This is a translation of Dmitri Pavlov's answer into a more intrinsic, more geometric, and more elementary language. In particular, I will show that the étale topos of a positive-dimensional variety is never the topos of a topological space. If $X$ is a topological space, then the associated topos $E = \mathbf{Sh}(X)$ is generated by subobjects of the final ...


1

You cannot extend the Steenrod squares to integer coefficients (as in have a set of cohomology operations satisfying the same axioms). Consider the tangent space $TS^2$ of the 2-sphere. With a little persistence, we can identify the Thom space of this vector bundle with the two cell complex $S^2 \cup_{2\eta} e^4$ (here $\eta$ is as usual the Hopf map ...


1

These are a few of the existence-type results that may be of interest to you. $\text{Green, Tao (2006)}$: There are arbitrarily long arithmetic progressions in primes. The proof they provided is based on extending Szemerédi's Theorem [1975, Endre Szemerédi] which, in itself, is an existence-type result, stating that any subset of $\mathbb{Z}$ with positive ...


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