The most efficient thermodynamics cycle is the Carnot cycle, it involves

1/ Receiving heat from hot source, expanding and doing work on the surround

2/ Receiving work from environment, being compressed and rejecting heat to surround.

So the efficient is limited by $n = \frac{(T\_H - T\_C)}{T\_H}$ where $T\_H$ and $T\_C$ are absolute temperature of hot and could source.

In the cycle, there is a step where heat engine receiving work from surround, thus lower the total work it can do. The question is, why does heat engine need that, why just ignore that, e.g. don't compress the gas so don't waste that work?

For example, let say we burn oil to heat air, the air is expanded and turn a turbine. We don't compress gas so any heat that air received will do work on the turbine.

By doing so, all received heat will be transfer to work, isn't it?

Why does every heat engine need to re compress gas to do the negative work on the environment?