K-theory. It's just papers citing constructions from other papers citing constructions from papers in French without any available translation. Wtf is anything. Try to explain it in layman's terms? uhhh.... if I could I think I would be published in a heartbeat. Nobody else seems to have such an explanation :(Maybe I could explain homotopy, and a splash of algebraic geometry, but this is not really beginning to cover the assignment of algebraic structures to all kinds of topological spaces that we call K-theory, it's only the beginning of homotopical and homological algebra, the setting in which K-theory reveals its necessity.

The layman's questions are: what is a hole? and who cares about a rigorous definition of a hole? Starting like 200 years ago a suitable definition of what constitutes a 'non-hole' was discovered. A 2-dimensional region contains no holes if you can draw a loop and guarantee that it can be shrunk to a point. A (number of) hole(s) is then a failure of this property. Similarly a 3 dimensional region contains no holes if you can paint a sphere and have essentially the same guarantee. This generalizes to higher dimensions, of course, as generalization to higher dimensions is any geometer's favorite thing to do.Now, who cares? Well the holes of a region determines some very nice properties of vector fields, specifically some properties that any physicist should care about. Holes of a region dictate how you can turn a crazy integral from vector calculus into a slightly less crazy integral from vector calculus using conservation laws (think Stokes theorem).

Now a splash of algebraic geometry: whether or not some loop \*can\* be shrunk to some other loop or some point, is, in general, a huge pain in the ass to determine sometimes. If instead of looking at such a geometric-topological property, we try to characterize this in terms of what functions exist over a region (be them complex functions, continuous functions, polynomial functions, or vector fields), we can turn something fundamentally useful, like conservation laws of vector fields, into a slightly less complicated question about polynomials for example, rather than directly looking at an underlying space. Basically we can use theories to establish Vector fields <-> geometric nonsense, and geometric nonsense <-> algebraic nonsense, to then cut out the middleman and be left with Vector fields <-> algebraic nonsense and somehow this actually cleans up some arguments quite a bit.

Now, K-theory? who fucking knows, man. Not me.