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 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.

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.

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 this 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.

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 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.