I have sent my manuscript to a professor of math about one month ago. This manuscript was written according to his encouragement for a joint research work. He told me that send him what I have done from my side, and then he will add additional materials from his own side. I want to send a tracking email about the status of that. Should I do that or I must wait for his reply?
"I want to send an email email asking him for an update about the status of that. Should I do that or I must wait for his reply?"
Absolutely. Why? Because it has been 1 month and you have not received any communication from him yet. Usually after waiting a full 7 days, it is not inappropriate to send a follow-up email.
"How much is an average time of waiting to recieve a reply from Professional co-author?"
It varies. Some people reply the same day, some the same hour, but it's also not uncommon for people to take a month or two to reply.
Usually it is not inappropriate to ask for an update 7 days after the first email. Another one after another 7 days can also often be considered okay. It's when you send more than 2 follow-up emails, or when you do not wait ~7 days between them, that it might start to be seen as "nagging", but even then that's not always the case. Please send him a follow-up email and ask us what to do if you still have problems after that. My guess is that he'll reply (since he asked you for the manuscript).
When communication becomes difficult, when human sensitivity comes into play, don't write mail after mail: call your collaborator by phone. Maybe on phone you can get them to make a credible promise; maybe they will make you understand how overburdened they are right now; maybe they will enjoy a little chat with you; hopefully, your relation with them will grow through direct conversation.