9:36 PM
@JonathanBeardsley mahowald proved that ko is not a thom spectrum in the following way: suppose ko = X^xi for some spherical bundle xi over X. then the thom isomorphism allows us to understand what the homology of X is; using this data, you show that a certain skeleton of X can't exist
similarly, he proved that ku is not an E_1-thom spectrum
rudyak generalized this to show that ku is not a thom spectrum (of any map X -> BGL_1(S)). his argument was different: suppose ku was a thom spectrum X^xi. let D denote the space classifying spherical bundles which are ku-orientable, i.e., the fiber of BGL_1(S) -> BGL_1(ku) = BGL_1(KU). this has a natural map to BGL_1(S), so you get a thom spectrum Mf, which is universal for ku-orientations
in particular, the identity on ku factors as ku -> Mf -> ku; by our knowledge of H_*(ku) and H_*(Mf), the latter of which i think was figured out by sullivan, you show that such a factorization is impossible on the level of homology
this gives a contradiction. note that this automatically gives the case of ko, since the wood cofiber sequence gives an equivalence ko /\ C(eta) = ku; if ko was a thom spectrum, then ku would be one too, since C(eta) is a thom spectrum
i also have a proof that BP<n> for n>0 is not an E_2-thom spectrum; the proof is again by contradiction: you assume otherwise, so BP<n> = Y^xi where Y = Omega^2 X for some X. then, you use our knowledge of H^* BP<n> to compute H^*(X), and show that some secondary power operation (due to adams for p=2 and liucevilius for p>2) leads to a contradiction
if you could strengthen this by showing that BP<n> for n>0 is never a thom spectrum, then an argument similar to what i wrote above would prove that tmf is not a thom spectrum either (since DA(1) is the thom spectrum of a spherical bundle over a finite complex)
i don't think that there is a standard way to show that something is not a thom spectrum
but in light of the above discussion, it seems reasonable to expect that something living at "finite chromatic height" (whatever that might mean) is never a thom spectrum. that's why i believe that k(n) for n>0 is not a thom spectrum.
oh, and there's also angeltveit-hill-lawson, which proves that ku and ko are not E_3-thom spectra by computing their thh and using the fact that thh plays nicely with thom spectra
that's the hardest method of proof for me, because i don't understand computations very well
it also seems to be really hard to guess that a (non-finite) spectrum is a thom spectrum (at least, mahowald's theorem on HF_2 seemed like black magic to me before i learnt that thom spectra give you free E_k-algebras on stuff)
