I don't see toning down the jargon as being advantageous to the common Joe, but it could be advantageous to mathematicians outside the extremely immediate field. For instance I'm guessing that any mathematician will understand "immersed surfaces in 4-manifolds" and maybe even "homology cylinder" even if the layman won't. However, how many mathematicians know "Whitney move" and "Whitney tower" and "Arf invariant" and "Milnor invariant" and "Sato-Levine invariant" and so forth? Some tiny fraction. A little less jargon could make your paper accessible to ten or a hundred times as many people.

Of course, quoting the abstract is a little unfair. The abstract needs to be short so there's no time to explain what the hell a Sato-Levine invariant is. Perhaps the actual paper is a lot clearer.

I am guessing that any mathematician will understand "immersed surfaces in 4-manifolds"?

I think you are guessing wrong. A 4-manifold, that I can more or less understand (and that is about as far as one can get, as there are several slightly different definitions in common use), but immersed surfaces in manifolds? And I do not think that is because I haven't done abstract math since graduating. One can graduate in mathematics without encountering any manifold, for example by focussing more on applied fields such as discrete math, analysis, or statistics.

I do not think that that abstract is necessarily bad, though. It seems to mention sufficiently many terms to sell it to the target audience of, say, 100 people world-wide. If you do not know any of these, you should read something more basic.