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The sphere of influence is a region around a supermassive black hole in which the gravitational potential of the black hole dominates the gravitational potential of the host galaxy(found it in wiki). So is it the same? If not, what holds the milky way together?

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No. The black hole at the centre of the Milky way is only a few million solar masses. The same amount of mass, made up mostly of stars, would be enclosed within $\sim 10$ parsecs of the Milky Way centre. The bulge of the Milky Way, which is a few thousand parsecs across contains more than 20 billion solar masses.

Thus the dynamics of our Galaxy, outside of the central few parsecs are hardly influenced by the black hole at all.

What holds the Milky Way together? The gravity of everything else - stars, gas, dust, dark matter (the latter probably being the most important once you get to galactocentric radii beyond the Sun).

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