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1 Easy Proposition Let $f:X\to Y$ be a continuous map of topological spaces, $\mathscr F$ a sheaf of abelian groups on $X$ such that $R^jf_*\mathscr F=0$ for $j>0$. Then for all $i\geq 0$ there exists a natural isomorphism $$ H^i(Y, f_*\mathscr F)\simeq H^i(X,\mathscr F) $$ Proof Apply the composition rule for the derived functors of $G=\Gamma(Y, \_ )$ ...


43

This is not really an answer, but rather a meta-answer as to why there exist many conventions in the first place. The symmetric monoidal category $\mathit{sVect}$ of super-vector spaces has a non-trivial involution $J$. The symmetric monoidal functor $J:\mathit{sVect}\to \mathit{sVect}$ is the identity at the level of objects and at the level of morphisms. ...


40

The category of sets admits precisely nine model category structures, no more no less. I learned this fact from Tom Goodwillie's comments on a different MO question. It always shocks people when I mention it to them, so I guess it is surprising. I am not sure which is more surprising, that you can actually compute all the model structures or that there are ...


38

This is a result in algebraic topology, where we study the structure of topological spaces $X$. One early way to do this is to calculate a thing called $H_*(X)$, the ordinary homology of $X$. Later people discovered various "extraordinary homologies", which give more precise information. There are very many different extraordinary homologies, including ...


37

It seems that the archives have been taken down because of "legal issues related to the ownership of the content." See http://publishing.mathforge.org/discussion/169/clockss-and-portico/#Item_18 for further discussion. Unfortunately, nobody involved seems to be publicly discussing the situation, and I have not yet been able to learn exactly what's going on....


37

Perhaps one of the biggest ideas that VV was pursuing is the Initiality Conjecture for Martin-Löf type theory (with universes). A rough idea is that from the rules for a type theory, one can define a category (the syntactic category) that has certain structure depending on which rules are admitted to the type theory. For instance, dependent products ...


36

I think Abdelmalek Abdesselam and William Stagner are completely correct in their interpretation of the words "Behind" and "one immutable source" as describing one theory, the theory of symmetric functions, being the central core of another. The issue that led to this question instead comes from misunderstanding this sentence: Today it is K-theory ...


31

If you are given a homotopy functor $L$ from spaces to spectra, the assembly map is a natural map of spectra $$ H_\bullet(X;L) \to L(X) , $$ where the domain is a homology theory. This homology theory is the "homology of $X$ with coefficients in $L$," that is, $X_+ \wedge L(\text{pt})$. This map is a universal approximation to $L$ on the left by a ...


30

Aside from his work on the foundations of mathematics, which others have already elaborated on, earlier in his career Voevodsky also proved the Milnor conjecture in algebraic geometry. The Milnor conjecture relates Milnor K-theory to Galois cohomology (or, equivalently, etale cohomology, since we are dealing with the case of a field). It is related to ...


27

Of course, in any spectral sequence $E_{r+1}$ is a subquotient of $E_r$ (the kernel of $d_r$ divided by the image of $d_r$). And in general new torsion can appear in the sense of torsion elements in $ker/im$ that are not represented by torsion elements in $ker$. But this cannot happen when the spectral sequence is rationally trivial, that is, when the ...


26

There is a very nice article by Peter May et al. which is all concerned with this question: H. Fausk, P. Hu, Peter May, Isomorphisms between left and right adjoints, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html) The article cleanly identifies three different flavors of ...


26

There is a basic way to see whether things like this should be true. Any bounded double complex of vector spaces over a field $k$ is (noncanonically) the direct sum of complexes of the following two sorts: Squares: $$\begin{matrix} k & \rightarrow & k \\ \uparrow & & \uparrow \\ k & \rightarrow & k \end{matrix}$$ Staircases:$$\...


26

I agree with Ryan that Serre's proof can be viewed as perfectly conceptual, but here is a modern version. Accept from Serre that the homotopy groups of spheres are finitely generated. Let $k\colon S^n \longrightarrow K(\mathbf{Z},n)$ be the canonical map. We know how to rationalize spaces and maps. The rationalization of $k$ is a map $k_{0}\colon S^n_{...


26

Here are some thoughts, gathered from reading many texts about algebraic K-theory. Let me start with some historical remarks, then try to give a more revisionist motivation of the plus construction. First of all, it's true as you say that the already-divined definitions of the lower K-groups made it seem like the higher K-groups, whatever they might be, ...


25

I was the only editor of the old "K-theory" who did not join the editorial board of the "Journal of K-theory". A decision I have never regretted. On the other hand, I do regret that it is no longer possible to access the old K-theory papers online. I do not believe the fault is with Springer. I urge all authors to make their papers available online. If you ...


24

I suspect the name just arose naturally (for obvious reasons) but that it would be tough to trace back to any single person. After Cartan-Eilenberg proved it in 1956 (Homological Algebra, p.40) the first mention I see in English is by Tate in 1966/67 (p-divisible groups, p.178) followed by Hartshorne in 1968 (Cohomological Dimension of Algebraic Varieties, p....


23

If $Y$ is a connected CW-complex of finite type which is both an H-space and a co-H-space, then $Y$ has the homotopy type of $S^1$, $S^3$, $S^7$ or a point. This is a result of Robert West: Robert W. West, $H$-spaces which are co-$H$-spaces, Proc. Amer. Math. Soc. 31 (1972), 580--582. It follows that if $X$ is a finite type CW-complex such that $\Sigma X$ ...


21

You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least the following properties K-theory of an affine scheme $\mathrm{Spec}\,R$ is given by the algebraic K-theory of projective $R$-modules in the sense of Quillen ...


20

Since for any two spectra $E,F$ we have $$ E_n(F)=[\mathbb{S}^n,E\wedge F]\cong [\mathbb{S}^n,F\wedge E] = F_n(E), $$ you may as well ask about $H_n(KU;\mathbb{Z})$, the integral homology of the complex $K$-theory spectrum. This calculation is carried out e.g. in Chapter 16 of the book Switzer, Robert M., Algebraic topology – homology and homotopy., ...


20

We have $KU_*(H\mathbb Z) = \pi_*(KU\wedge H\mathbb Z)$; this ring is concentrated in even dimensions and carries an isomorphism between the additive and multiplicative formal group law, hence is rational. Thus the map $$ KU\wedge H\mathbb Z\to (KU\wedge H\mathbb Z)\wedge H\mathbb Q\cong KU\wedge H\mathbb Q $$ is an isomorphism. The Chern character $KU\to \...


19

I don't really know if this helps, but you can in effect give the plus-construction definition of $K$-groups without explicitly mentioning homotopy groups, and without ever doing the plus construction: $H_1BGL(R)$ is $K_1(R)$. Map $BGL(R)$ to an Eilenberg-MacLane space $BK_1(R)$ and consider the homotopy fiber. This space has trivial $H_1$. Its $H_2$ is $...


18

Concerning the definition of symplectic K-theory: there are various possible definitions, the homotopy groups of the plus-construction of the classifying space of the infinite symplectic group is one such definition. Other definitions can be given using categories of forms (similar to Q-construction or $S^{-1}S$-construction for algebraic K-theory) or a ...


18

I think you are misinterpreting the quote. In the last sentence, the word "source" does not mean "source of these theories (K-theory, categories, group representations", but "source of the theory of symmetric functions". Rota is not claiming that K-theory, etc. have "at their core the ordinary, crude definitions of symmetric functions", but that "the theory ...


17

I think I have been able to reproduce the "argument by Wagoner" (perhaps it was removed from the published version?). It certainly holds in more generality that what I have written below, using the notion of "direct sum group" in Wagoner's paper (which unfortunately seems to be a little mangled). Let $M$ be a homotopy commutative topological monoid with $\...


17

You can refine this. Let's take $k=2$ to give the idea. To a based set $X$ you can associate $(X\wedge X)/X$, a based set with free action of $\Sigma_2$. This leads to an operation going from stable homotopy of $S^0$ to stable homotopy of $B\Sigma_2$, such that when followed by transfer it gives the difference between the identity and squaring. This leads ...


17

The operation which sends a finite set $S$ to its set of $k$-element subsets, $\binom{S}{k}$, gives rise to the $k$-th stable Hopf invariant. There is additional structure in this: the set $\binom{S}{k}$ has a canonical $k$-fold covering so the operation is better viewed as a map $$ QS^0 \to Q(B\Sigma_k)_+ $$ rather than as a map $$ Q S^0 \to QS^0 , $$ ...


17

Thanks to a librarian's listserv here an update: Since recently K-Theory is now available through Portico (only available to members) - see http://www.portico.org/digital-preservation/news-events/news/general-news/portico-now-provides-access-to-k-theory-content But no news if there are any plans to change the situation at the Springer website...


17

Here is one conceptual description of the relationship. (I wouldn't call it an explanation; I don't really know why it's true, except that it's because Bott periodicity is true.) KO-theory is the first Weiss-derivative of the $K$-theory of Clifford algebras. More precisely: given a real inner product space $V$, we get a category $M(V)=\mathrm{Mod}(Cl(V))$ ...


17

This answer probably won't be coming from the perspective that you want, since it'll use even more stable homotopy theory than Adams did. But I think it's pretty clear in its own way. First let me make a small correction to Neil's comment. Actually, from the J-homomorphism construction you only get a map of spectra from the connective cover of $\Sigma^{-1}...


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