Analogy behind Hyman Bass' definition of algebraic $K_1$

Question

On pp.78 of these notes live TEX-ed by Arun Debray for Dan Freed's K-theory course (lecture 23 given by Andrew Blumberg), there is a comment about how Hyman Bass initially started from the topological $K_1$ to define the algebraic $K_1$ as the Grothendieck group of the automorphism category. It claims that his definition was based on the analogy of the suspension and clutching functions from topology. I really cannot see this analogy and how it plays a role in the categorical definition of algebraic $K_1$ in the notes.

I think it would be very useful to know the analogy between topological and algebraic $K$-theory. Thus could somebody clarify the analogy and its role in the definition? In addition is there any attempt of generalizing this analogy to $K_i$ with $i > 1$ (despite a natural definition by Quillen exists) and why it fails?

Answer 1

For a space $X$, isomorphism classes of rank-$k$ vector bundles on the suspension $SX$ are in one-one correspondence with homotopy classes of maps from $X$ to $GL_k(F)$, where $F$ is ${\mathbb R}$ or ${\mathbb C}$. You get $K_1(X)$ by replacing $GL_k(F)$ with the direct limit $GL(F)$. The algebraic analogue, then, is that $K_1(R)$ should be generated by starting with "homotopy classes" of maps from $Spec(R)$ to $GL_k$, or equivalently elements of $GL_k(R)$, and then taking a direct limit. So you want elements of $GL(R)$ modulo "homotopy".

There's an obvious sense in which elementary matrices are homotopic to the identity (given a matrix of the form $I+M$ where $I$ is the identity and $M$ has one non-zero element, consider $I+tM$), so it's natural to try modding out by these.

The Karoubi-Villamayor construction is a sort-of-generalization to higher $K_i$ but it only works well when $R$ is regular.