Added a description of the ordinary fiberwise integration of differential forms.

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in terms of pushing by dual morphisms in KK-theory.

]]>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..)

]]>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 $H^*(E,\mathbb{Z})\to H^{*+k+1}(E,\mathbb{Z})$.

]]>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)

]]>But this is the classical case, isn't there a generalized version ?

]]>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.

]]>So what is "Thom spectrum" ?

]]>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).

]]>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.

]]>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 :-).

]]>started fiber integration

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