# Tagged Questions

**1**

vote

**1**answer

223 views

### Proof that the Hodge-de Rham Rank Equals the Euler Characteristic

Can someone please provide a good (online accessible) reference for the well-known identity
$$
\text{rank((d + d}^*)^+) = \sum_{i=}^n (-1)^i \dim(H^i(M)),
$$
where $M$ is a manifold of dimension $n$, ...

**3**

votes

**0**answers

168 views

### The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:
Let $j: ...

**4**

votes

**1**answer

596 views

### The equivariant index of Dirac operator

Let us consider the Dirac complex
\begin{equation}
D_{\rm Dirac}:S^+\to S^-
\end{equation}
where $S^{\pm}$ are the chiral-spinor bundles on $\mathbb{R}^4$.
Using the fact that the bundle $S^+$ is ...