#### Alladi Ramakrishnan Hall

#### Zero cycles, Mennicke symbols and $\mathrm{K}_1$-stability of real affine algebras

#### Sourjya Banerjee

##### IMSc, Chennai

*Let $R$ be an affine algebra of (Krull) dimension $d \geq 2$ over the *

base field $\mathbb{R}$. In this talk, we shall discuss the following:

First, we define the $d$-th weak Euler class group

$\mathrm{E_0^d}(R)$, as defined by Bhatwadekar-R. Sridharan, the

Levine-Weibel Chow group of zero cycles

$\mathrm{CH}_0(\text{Spec}(R))$ modulo rational equivalence, and a

canonical map between them. Then we show that for a certain class of

real algebras, this map is an isomorphism. In the second part, we

define the weak universal Mennicke symbol and the universal Mennicke

symbol of length $d+1$ over $R$. Further, we show that these coincide

for the same class of real algebras. If time permits, then we shall

discuss some stability results for the groups $\mathrm{K_1}(R)$ and

$\mathrm{K_1Sp}(R)$.

Done