Connect 2 pins or set a pin to the ground

Hello !

I want my Arduino to take photos with my Canon EOS 600D, I bought an external button to take the photos and I cutted the wire just before the button itself. There is 3 wires : White => When sent to Ground, it takes a photo Red => When sent to Ground, it set the focus if the lens is in auto-focus mode and the Ground wire.

Here is a little photo :

I would like to put the white wire in a pin of my Arduino and either find a program solution to make this pin a Ground when I want to take a photo, or put the ground wire on another pin of my Arduino and connect the 2 pins when I want.

Is there a way to do this without buying other electronic components ?

Thank you in advance

Well if it was me I would buy a small relay with 2 separated pins and connect then on the N.O(Normally Open) side and with a small transistor connected to an arduino pin do the switch safely.This way you will separate the systems and voltage differences. Then In arduino program all you have to do is set the pin low or High. The extra hardware is cheap (relay, transistor,Diode and some resistors) I don't think you can achieve this without a small circuit.