Homology as Abelianization

2009-05-31T21:20:00.002+04:00

Passing from a topological space to its homology groups is something like abelianization. For example H₁=π₁/[π₁,π₁]. But this principle makes sense even literally: (as follows from Dold–Thom theorem) {homology groups of X} = homotopy type of Z[X] — topological abelian group freely generated by points of X.

To get standard definition one should pass from topological space to simpicial set before applying Z[-]. Canonical way to do so is to apply functor (-)_Δ adjoint to geometric realization (that is [n]->C(Simp[n];X) where Simp[n] is standard n-dimensional symplex). Resulting simpicial abelian group Z[X_Δ] can be (by Dold–Kahn equivalence) considered as a complex; and this is the singular complex of X. This, BTW, explains why Dold–Thom theorem is obvious (modulo Dold–Kahn theory).

What's wrong with this picture is complete lack of connection with standard framework of spectra. Of course, now one can recover Eilenberg–Maclane spectrum as Z[Sⁿ]. But looking only on the spectrum it's unclear (to me, at least) why (for the corresponding homology theory) there exists dg-lifting (aka chain complex).

The test question here is to find analogue of this picture for some extraordinary (co)homology theory. Probably the simplest case is complex K-theory. What is the analogue of singular complex for K-theory? (E.g. what is algebraic description of stable homotopy category Bousfield-localized in KU?)

Orientation puzzle(s)

2008-10-22T00:14:00.004+04:00

Let me begin with a fill-in-the-gap puzzle.

C^× ~ S^1	BS¹ aka K(Z,2) aka CP^∞	complex orientability of a homology theory: h(CP^∞)=h(pt)[[u]] (deg u=2); example: H(-;Z)	complex vector bundles, Chern classes	complex K-theory, BU	unitary cobordism, MU
R^× ~ Z/2Z	B(Z/2Z) aka K(Z/2,1) aka RP^∞	real orientability of a homology theory: h(RP^∞)=h(pt)[[t]] (deg t=1); example: H(-;Z/2Z)	real vector bundles, Stiefel-Whitney classes	real K-theory, BO	unoriented cobordism, MO
Z/pZ (p>2)	B(Z/pZ) aka K(Z/pZ,1) aka L_p	“Z/pZ-orientability” of a homology theory: h(L_p)=h(pt)[[ε,u]] (deg ε=1, deg u=2; &epsilon²=0); example: H(-;Z/pZ)	???	???	??? (universal Z/pZ-oriented theory)

Well, at least in the complex case (the first line of the table) it's known how to construct corresponding homology theories starting from the classifying space: consider Σ^∞CP^∞ and invert a generator of π₂CP^∞; by Snaith's (?) theorem resulting space is homotopy equivalent to BU. And applying the same procedure one more time yields unitary cobordism: Σ^∞BU[1/β]=\wedge_nMU(n). I wonder, if there is a variant of that construction which works in the real case? N.B. inverting generator of π₁ in Σ^∞RP^∞ or Σ^∞BO yields a contractible space.

And the last question (in this post, I mean :-). Consider R:=H^*(B(Z/p);Z/p)=Z/pZ[[ε,u]]. Structure of H-space on B(Z/p) induces on Spec(R) structure of an algebraic group — and in fact it's just A^1|1 (a topologist would probably say something like "structure of 1|1-dimensional additive formal group on Z/pZ"). Theorem: Z/pZ[Aut Spec(R)]=A_p, where A_p is the automorphisms algebra of the homology functor (a topologist would say "algebra of mod p cohomological operations") aka mod p Steenrod algebra. Does anyone know an explanation of this theorem?

Magia Mathematica

Homology as Abelianization

Orientation puzzle(s)