What holds a violin's soundpost in place?

Friction. The soundpost is a small spruce dowel wedged vertically between the front and back plates inside the body, and it is not glued to either - it is held purely by the pressure of the strings pushing the top down onto it. It can be knocked over if the strings are all slackened at once, which is why luthiers loosen them one at a time.

It does two jobs. It stops the thin spruce top collapsing under about 25 kilograms of string tension bearing down through the bridge, and it transmits vibration to the back plate so the whole box resonates rather than just the front. Together with the bass bar glued under the top on the other side, it is what makes the instrument loud enough to be heard.

Its position is critical to a degree that sounds like superstition and is not. Moving it a millimetre audibly changes the tone, and setting it is a specialist skill done with a long hooked tool through an f-hole, by feel. The Italian name is anima - the soul - and the French is the same. That an instrument's character is decided by an unglued stick nobody can see is a fair summary of why violin-making resisted being systematised for so long.