noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
For a given manifold $X$ of finite dimension there exists an embedding $i : X \to \mathbb{R}^n$ into some Cartesian space. Using the Pontrjagin-Thom collapse map this induces a morphism in topological K-theory
Similarly for any point inclusion $j : * \to \mathbb{R}^n$ there is such a morphism $j_! : \mathbb{Z} = K(*) \to K(T \mathbb{R}^n)$ which is an isomorphism – the Thom isomorphism.
The topological index of topological K-theory on $X$ is the composite
One can prove that this is independent of all the occurring choices. In particular it does not depend on the specific choice of embedding of the manifold $X$ into to the Euclidean space. The topological index function is uniquely fixed by two properties (this is the content of the Atiyah-Singer index theorem):
For $X$ a point we have $ind_t=id$.
Index functions commute with the maps $i_!$.
From this one defines the topological index of an elliptic operator . The principal symbol of the operator defines a homogenous length-one chain complex of bundles on $T X$ exact outside the null section. Elements of this kind are precisely cycles for the compactly supported K-theory of $T X$ hence an elliptic operator $D$ has a topological index only depending on its principal symbol.
On the other hand, analysis associates to $D$ its analytical index that is
on one (hence all) Sobolev space that $D$ is defined on.
The Atiyah-Singer index theorem states that the analytical index of $D$ is equal to its topological index.
The story starts with an embedding $i:X\to Y$ of compact manifolds. In this situation one can construct a homomorphism
between the compactly supported K-theories of their tangent bundles.
Notice here the reverse functoriality: for the base space $K$ is contravariant while for the total spaces of the tangent bundles it is covariant. This uses the Thom mapping: if $X$ is a compact manifold and $V$ a real vector bundle over $X$ there is a natural map
One of the most important results of K-theory, namely Bott periodicity, can be seen as the statement of the fact that this map is an isomorphism. Now apply this construction to the normal bundle $N$ of $X$ in $Y$ to get
and (looking at $N$ as a tubular neighbourhood of $X$ in $Y$) compose it with the natural map
to get $i_!$.
Now given a manifold $X$, embed it in a Euclidean space $\mathbb{R}^n$ for some suitable $n$ and consider the inclusion $\{0\}\to \mathbb{R}^n$. This induces the (Thom isomorphism) mapping $j_!:\mathbb{Z}=K(\{0\}) \longrightarrow K(T\mathbb{R}^n)$.
The topological index is defined to be
topological index