What advantages and disadvantages should I consider in deciding whether to publish my academic book traditionally or self-publish? I am an amateur mathematician. I am writing a research monograph in the field of abstract mathematics (general topology, specifically).

Should I publish it traditionally or self-publish?

There are many benefits of self-publishing (e.g with Lulu) an academic work:


No need to tremble awaiting my book to be rejected by a peer review. No responses like "first publish in articles, only then make a book". I am in full control what I want to publish.
I don't pay the publisher 80-90% of my revenue. (This also may make AdWords marketing of my book profitable and thus I can do a rather huge advertisement of my book myself using AdWords. I suspect, this may over-perform traditional publishers in the number of sales.)
I am in a full control of my book. No forced need to change something, if an editor's opinion differs from my own.
No need to convert it to LaTeX, I can use my preferred software such as TeXmacs.
The book goes into Amazon and other distribution channels anyway.


I can pay a professional scientific editor to edit my book for paid, as a kind of business investment.

Peer review is intended to choose which books are published and which are not. I can do fine without peer review, allowing the buyers of my book to decide for themselves.

Well, a potential buyer may prefer books published by a big publisher, but it (in my opinion) can be well replaced with big red letters "Edited and checked for errors by professor XXX."

Drawbacks which I know:


It may not be as good for my academic carrier as traditional academic publishing. (It does not matter for me anyway, as I am not a professional academic.)
Not sure if my book goes into university libraries (please comment on this issue).


I've pointed many benefits of self-publishing. What are drawbacks (except of pointed by me)?



And one more specific question: Is the number of books sold if using a traditional publisher, likely to greatly overperform the number of books sold if self-publishing? If yes, why?
Should I publish it traditionally or self-publish?


Why do you want to publish at all?  You answered


  I write the book to store down my research results and to spread my new knowledge. To make money is not the main aim, but it would be nice.


Given that: the answer is that you should certainly not self-publish your work.  You can store your results and spread knowledge by having the material freely available on the internet, as I believe is already the case.  The arxiv is one nice place to put work, but it is not the only one: you could put in on github or any number of other repositories.  You can just put it on your own website and make sure that google indexes it.  That means that billions of people can access it at any time.

Let me be clear with you: you are not going to make money self-publishing works of mathematics that you have not been able to publish traditionally.  It is exceedingly rare for any mathematical text beyond the undergraduate level to make a profit that is worth the time taken to write it.  (Maybe a few of Serge Lang's books qualify; probably not.)  If you go self-publishing rather than traditional publishing, you will lose money, and what you're paying for is the vanity of being a published author. 

The bar for interest by the mathematical community is much lower than the bar for the type of public interest needed to generate any real sales.  The thought that you have "My ideas are too bold for the mathematical community, so I need to take matters into my own hands; they don't know the value of my work as well as I do" is not only crankish but actually specifically damaging to you: it makes you ideal prey for predators of various kinds.  You told us in a previous question that you literally fell prey to a diploma mill and thereby lost money.  The same mindset that you have now is going to cost you more money in the future.  

I'm sorry to tell you this, but this has been going on for several years now, so I feel I should be plain: no one in the world has found your work to be of significant mathematical value.  This means that, with probability slightly less than one, that your work does not have significant mathematical value.  But in the unlikely event that your work does have value, you're not doing what is necessary in order to show it.  Mathematical research is not about simply writing down structures that generalize other structures and proving results about them.  You have to solve old problems or pose new ones that are of interest to the community. Bold statements of superiority would be a positive thing if they are specific and factual: for your work to be "superior", it should solve at least one problem that others have posed.  If you've done that, please explain yourself properly and then your work can be published in the mathematical mainstream.  If you haven't: please start to be honest with yourself about the value of your work.  Your livelihood is at stake.