As Tilman said, a Pontrjagin-Thom construction can be used to define umkehr maps in all generalized cohomology theories $E$, provided that the stable normal bundle is $E$-oriented.
Sufficient conditions for the existence of PT-constructions are:
$f:X \to Y$ is a proper map of (possibly noncompact) manifolds.
$f:X \to Y$ is a bundle with fibre $M$ and structure group $Diff(M)$. $M$ needs to be a closed manifold.
One can lax the smoothness assumption considerably, and the appropriate tool in this situation is the Leray-Serre spectral sequence. Assume that $f:X \to Y$ is a fibration and the fibres are homotopy equivalent to closed oriented $n$-manifolds. Assume moreover (for simplicity) that the local coefficient system $H^n (X/Y)$ (by which I mean the $n$th cohomology of the fibres) is constant and that the fibres are connected. Then you get a map $$H^{n+k}(X) \to E^{k,nE_\infty^{k,n} {\infty} \subset E^{k,nE^{k,n}_{2} {2} = H^k (Y)$$H^k(Y)$$ and this is your wrong-way map.
The most elegant way to show that all the different constructions coincide in their common domain of definition (smooth bundles over manifolds), it might be best to characterize the wrong-way maps by a short list of axioms:
- $f_! : H^{\ast}(X) \to H^{\ast}(Y)$ is a homomorphism of $H^{\ast}(Y)$-modules
- $f_! $ is natural with respect to pullbacks of bundles
- $f_!$ is normalized, i.e. if $Y$ is a point, then $f_!$ is evaluation against the fundamental class.
It is not too difficult to show that these axioms force $f_!$ to coincide with the spectral-sequence definition.
