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started fiber integration
added a few more details and a set of slides as a reference.
The big question for me is: what is a more abstract way to think of that integration map $\int_F$ ? It looks a bit wild. There must be a way to say this more elegantly, I’d hope.
Thursday I asked somebody who is expert on this stuff for a good reference on the general theory of fiber integration in generalized cohomology. He just shrugged :-).
This pushforward is either with compact supports or alike. I always emphasise proper pushforward in similar business, and is different from just pushforward. In algebraic geometry it is usually much more studied. References in algebraic topology I do not know but it is related to Spanier-Whitehead duality business and Thom spaces, so probably there is a good treatment in Rudyak's book, but have no time to check now.
I found this reference here
which seems to go a long way towards the goal of reducing the fiber integration formula to abstract nonsense.
Effectively what they say is that one should look at functors $Top^{op} \to Spectra$, represented by spectra by sending a spectrum not just to its cohomology on each given space, but to its mapping spectrum . Then notably the construction of the Thom space of the (stable) normal bundle of some $P$ – crucial in the fiber integration business – is just, they say, the mapping spectrum $hom(P, S^0)$, where $S^0$ is the sphere spectrum (page 5).
They discuss fiber integration in deRham cohomology in this language (page 6).
So what is "Thom spectrum" ?
So what is “Thom spectrum” ?
Its the spectrum given by the sequence of Thom spaces $M SO(n)$ associated to the associated bundle of the universal $SO(n)$-bundle for all $n$.
Have to run now. More later.
But this is the classical case, isn't there a generalized version ?
I’m not completely convinced of the abstract nonense approach in Cohen-Klein. in a sense, it can be summarized by saying that a wrong way map is induced by a right way map and a duality. and this duality explicitly calls in a notion of sphere bundle associated to a virtual vector bundle, so that in many ways this abstract nonsense is much less abstract than it would like to be.
(I have a suspect this is the same comment as Zoran’s, only written in different words)
something closely related, and which we seem to be missing an nPOV on (and an nLab entry) is Leray-Serre spectral sequence of a fibration $F\to E\to B$.
in the particular case of a sphere bundle on a connected base, the $k$-th differential in the spectral sequence $d_k:\mathbb{Z}=H^0(B,H^k(S^k,\mathbb{Z})\to H^{k+1}(B,H^0(S^k,\mathbb{Z})$ determines a distinguished element $e:=d_k(1)$ in $H^{k+1}(B,\mathbb{Z})$, the Euler class of the sphere bundle, and integration along the fibres fits into an exact triangle whose other two edges are the pullback $H^*(B,\mathbb{Z})\to H^*(E,\mathbb{Z})$ and the cup product with the Euler class $H^*(E,\mathbb{Z})\to H^{*+k+1}(E,\mathbb{Z})$.
just a toy exaple for today. assume we want to compute the cohomology group $H^n(X\times S^k,A)$ for some abelian group $A$. if $\mathbf{H}=\mathbf{Top}$ is the oo-topos of nice topological spaces, then what we want to determine is $\pi_0\mathbf{H}(X\times S^k,\mathcal{B}^n A)=\pi_0\mathbf{H}(X,\mathbf{H}(S^k,\mathcal{B}^n A))$. so we are interested into maps with target $\mathbf{H}(S^k,\mathcal{B}^n A)$. as we have seen in comment 119 here, this space has at most two nontrivial homotopy groups (exactly one if $n\lt k$ and exactly two if $n\geq k$): $\pi_n\mathbf{H}(S^k,\mathcal{B}^n A)=A$ and $\pi_{n-k}\mathbf{H}(S^k,\mathcal{B}^n A)=A$. so, if $n\lt k$, then $\mathbf{H}(S^k,\mathcal{B}^n A)\simeq \mathcal{B}^n A$, so we recover that $H^n(X\times S^k,A)=H^n(X,A)$ for $n\lt k$. in the $n\leq k$ situation, the Postnikov tower of $\mathbf{H}(S^k,\mathcal{B}^n A)$ involves a $\mathcal{B}^n A$ and a $\mathcal{B}^{n-k} A$, and we recover that $H^n(X\times S^k,A)$ has a contribution from $H^n(X,A)$ and one from $H^{n-k}(X,A)$.
(I have to expand this, but have to run now..)
added a section
in terms of pushing by dual morphisms in KK-theory.
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