The simplest and best answer may be beneath.
But it's complicated:
First, let's explore atop, which has the meaning of "directly on top of," as in this example:
The house sits atop a hill.
A quick search through concordances (visit www.lextutor.ca to find good concordance tools) revealed that atop is most commonly used for outdoor structures (as in the example) situated in the terrain and for the structures on top of outdoor structures (weather vanes atop barns & air raid sirens atop court houses, to cite a couple of examples that you will find simply by googling "atop"). So you could, if you wanted to, say The book is atop the table with the reasonable expectation that people will understand you, but on the whole "atop" is not a very common word. That said, if you punch "atop" into Google Ngrams, you'll find a gentle rise in occurrence from 1920 to 1975, followed by a relatively steep increase in usage from 1975 to 2000 (interesting to that there is a spike... or more like a hill... in the early 40s). But perhaps I digress.
More to the point: what would an opposite to atop actually mean? If the house sits atop the hill, then would we speculate what lies... abottom the hill?
You might say: well, the house was built on stilts because the homeowner wanted (for whatever reason) to ensure good air circulation ____________ the house. In this case, I think the word beneath is the best fit. And if you look at different definitions of "beneath," you will find that most use the word "directly" or the phrase "in close contact." So that may qualify "beneath" as the opposite of "atop." However, if I wanted to talk about air circulation "above" the house, I'd say just that, not "atop." Because "atop" includes the idea of "attached to" or "sitting directly on" and is used for concrete physical objects, a category that "air" doesn't fit into.
So, we're left looking for a preposition that includes that meaning of direct contact, but for the bottom of things not the top. The predominant image in my mind is that of a barnacle, clinging to the underside of a boat. But I don't think we have a single preposition to describe that clinging simply because of gravity. All of the things that sit "atop" other things do so because doing so obeys the law of gravity. But clinging to the underside of something is simply not common enough, physically, to warrant a preposition.
Now, we haven't entertained the word below as a candidate. "Below," however, indicates a general directional position and does not include the idea of "directly" or "in close contact." The opposite would be "above," which most would agree does not include the same "direct contact" sense of "atop."