nLab
topological index

Idea

For a given manifold $X$ of finite dimension there exists an embedding $i:X\to {ℝ}^n$ into some Cartesian space. Using the Pontrjagin-Thom collapse map this induces a morphism in topological K-theory

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

Similarly for any point inclusion $j:*\to {ℝ}^n$ there is such a morphism $j_{!}:\mathbb{Z}=K(*)\to K(T{ℝ}^n)$ which is an isomorphism – the Thom isomorphism.

The topological index of topological K-theory on $X$ is the composite

$${\mathrm{ind}}_{\mathrm{top}}:\mathbb{Z}=K(TX)\stackrel{i_{!}}{\to }K(T{ℝ}^n)\stackrel{j_{!}^{-1}}{\to }K(*)=\mathbb{Z}.$$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 ${\mathrm{ind}}_t=\mathrm{id}$.

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 $TX$ exact outside the null section. Elements of this kind are precisely cycles for the compactly supported K-theory of $TX$ 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

$$dimKer(D)-dimcoKer(D)$$\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

$$i_{!}:K(TX)⟶K(TY)$$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

$$\phi :K(X)⟶K(V).$$\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

$$\phi :K(TX)⟶K(TN)$$\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

$$K_{*}:K(TN)⟶K(TY)$$K_* : K(T N)\longrightarrow K(T Y)

to get $i_{!}$.

Now given a manifold $X$, embed it in a Euclidean space ${ℝ}^n$ for some suitable $n$ and consider the inclusion $\{0\}\to {ℝ}^n$. This induces the (Thom isomorphism) mapping $j_{!}:\mathbb{Z}=K(\{0\})⟶K(T{ℝ}^n)$.

The topological index is defined to be

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

Related concepts

• index theory

  • topological index

  • analytical index

  • Atiyah-Singer index theorem

• eta invariant

• fiber integration in differential K-theory