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One standard notational convention is that (2) = (E) for unbased spaces $X$, where the brackets mean unbased homotopy classes of maps. This is consistent with the classification theorem that equiva …

Tom, not precisely sure what you want. I'm guessing you want to think of a commutative symmetric ring spectrum $R$ and then an $R$-algebra $A$. Without the extra layer, just using an $S$-algebra $R$ …

One answer these days is to think of $K$-theory as represented by a commutative ring spectrum (alias $E_{\infty}$-ring spectrum) $K$. Then there is a perfectly good theory of $K$-module spectra, to w …

There is a beautiful observation of my advisor John Moore that to my mind ought to be part of the focus of any such argument: the Hopf algebra $H_{\ast}(BU;Z)$, or equivalently $H^*(BU;Z)$ is self-dua …

Jacobo, I don't know where to put answers to questions asked after answers, and I don't have much of an answer. I do know how to manufacture the localization of $BSF$ at a prime $p$ out of symmetric …

An evident general construction is to take any multiplicative cohomology theory $E^*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bund …

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 …

You can find an elementary categorical comparison of different contexts of the sort you ask about in a paper by Halvard Fausk, Po Hu, and myself, entitled ``Isomorphisms between left and right adjoint …

Perhaps my very short (4 pages plus bibliography) paper ``A note on the splitting principle'' http://www.math.uchicago.edu/~may/PAPERS/Split.pdf may be illuminating. It shows that the splitting princi …