nLab topological index

Idea

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

$$i_! : K(T X) \to K(T \mathbb{R}^n) \,.$$

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

$$ind_{top} : \mathbb{Z} = K(T X) \stackrel{i_!}{\to} K(T \mathbb{R}^n) \stackrel{j_!^{-1}}{\to} K(*) = \mathbb{Z} \,.$$

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):

1. For $X$ a point we have $ind_t=id$.

2. 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

$$\operatorname{dim}\operatorname{Ker}(D)-\operatorname{dim}\operatorname{coKer}(D)$$

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.

More on the Thom map

The story starts with an embedding $i:X\to Y$ of compact manifolds. In this situation one can construct a homomorphism

$$i_{!} : K(T X)\longrightarrow K(T Y)$$

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

$$\varphi:K(X)\longrightarrow K(V) \,.$$

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

$$\varphi : K(T X)\longrightarrow K(T N)$$

and (looking at $N$ as a tubular neighbourhood of $X$ in $Y$) compose it with the natural map

$$K_* : K(T N)\longrightarrow K(T Y)$$

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

$$ind_t := j_!^{-1}\circ i_! \,.$$

Related concepts

• index theory

  • topological index

  • analytical index

  • Atiyah-Singer index theorem

• eta invariant, zeta function of an elliptic differential operator, functional determinant

• fiber integration in differential K-theory