Hi Frederir,
You can charge lithium-ion or lithium-polymer batteries with the Intel® Edison kit for Arduino* and the Intel® Edison breakout board. The Intel® Edison kit for Arduino* has a 100 mA charging current; the Intel® Edison breakout board has a 190 mA charging current. However, both use a charger IC.
You can check these documents where there is more information about this boards. There are some important details about which type of batteries you should use with the Edison:
Edison Breakout Board Hardware Guide
Edison Arduino expansion board hardware guide
Let us know if this information answered your question. Have a nice day!
Regards,
Diego.