How can we get the manganese (IV) oxide back after the reaction?

I assume you are referring to the decomposition reaction of hydrogen peroxide, in which manganese (IV) oxide is used as a catalyst. Hydrogen peroxide decomposes to form water and oxygen gas, according to the following chemical reaction:

`2H_2O_2 -> 2H_2O + O_2`

Manganese (IV) oxide is a catalyst and does not react with hydrogen peroxide or any of the reaction products. It is insoluble in water and only speeds up the rate of reaction (which is the primary function of the catalysts). 

The oxygen gas generated in this decomposition reaction escapes into the atmosphere, leaving only water and the catalyst in the reaction vessel (beaker, conical flask, etc). 

Since, manganese (IV) oxide is insoluble in water and has a very high melting point (~ 550 degrees Celsius), it can be separated by boiling the water and evaporating it, leaving solid manganese (IV) oxide behind.

That is how, we can reclaim the catalyst (manganese (IV) oxide) after the decomposition reaction has taken place.

Hope this helps.