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In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see section 2.2 (page 16) in Landweber - K-theory and elliptic operators article for more details. I would like to understand

why this construction is functorial, i.e. why $(j \circ i)_!=j_! \circ i_!$ where $i:X \to Y$ and $j:Y \to Z$ are embeddings of compact manifolds.

Forgive me if this question is too elementary for this site.

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  • $\begingroup$ Could you point to a more specific place in the article? The string 'wrong' does not seem to occur in it. $\endgroup$
    – LSpice
    Jul 1, 2018 at 16:16

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You will find a careful proof of this in Eldon Dyer's monograph "Cohomology theories". It is true for any multiplicative cohomology theory $h^*$ and smooth maps $i:X\to Y$ and $j:Y\to Z$ which are $h^*$-oriented, meaning with a choice of Thom class for their stable normal bundles.

The definition of the Umkehr is on page 53, and the functoriality result is stated as Theorem 8 on page 57.

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  • $\begingroup$ Thank you for pointing me this reference. However the proof in this generality requires a lot of machinery-I would be glad if it can be done directly (which I believe is the case since in Landwebers note all constructions are rather concrete) $\endgroup$
    – truebaran
    Jul 15, 2018 at 22:28

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