It's totally safe to install Viewer, since the Viewer downloads page provides specific SHA checksums you can use to verify with 100% certainty there's not been any virus injected into the Viewer package.
Knowledge is strength. Learning how to use precise security measures like SHA is far safer than putting blind faith into an imprecise package like Avast, which doesn't know anything about Viewer specifically but instead makes guesses based on algorithms that often, but not always, may catch a threat, but at the cost of many false positives.
Why is Avast identifying this virus as being in the Manifold Viewer.
First, because Avast doesn't have the resources or technology to rapidly enough scan many files and to identify 100% of threats without reporting false positives. It's extremely expensive work to keep track of millions of viruses and applications with enough diligence to protect against viruses without generating false positives. Even if Avast had the resources to do that, they would have to sell their product for such high prices that nobody would buy it. So their business model assumes they'll miss some viruses and also that they'll falsely libel some clean applications.
Second, because Avast doesn't have to pay any penalty when they libel another company's products. In theory they can't commit that crime without punishment (it is, indeed, a crime and not just a civil matter in many jurisdictions) but in practice their business model is based on getting away with it.
In theory, Manifold could sue Avast for libel, tortuous interference, and other causes of action arising from Avast's falsely saying that Viewer contains a virus. In theory, Manifold could also file criminal complaints in jurisdictions where Avast's libel is a crime and not just a civil matter or a tort (a "tort" in law is a civil crime). In wildly absurd theory, the police and prosecutors in such jurisdictions might actually do something about that, enforcing their laws.
Think about it: if you started telling people that a bank in your city employed criminals as tellers, and so if you did business with them those criminals could steal money from you, that would be libel. In almost all civilized countries, the bank could sue you. In many countries, if you were making money from such libel, say, by selling a report that alleged to save people from "infected" banks, it would be a crime as well.
But legal systems are so broken in so many countries that as a practical matter it is a waste of time and not worth it economically for any of Avast's victims to pursue the matter in court. That's especially true given that courts seem to have a blind spot in their common sense when it comes to anything having to do with computers. So if you stand on a street corner and libel a bank, they get that, but if you do the same thing on a web site or in an application, that confuses them, running up against that part of their brain where anything more technical than 1 + 1 = 2 is a cosmic mystery to be avoided.
So, for example, to bring a case against Avast you'd have to find some plaintiffs with experience of Avast and the specific libel it commits against Viewer, file a case in that particular jurisdiction (means retaining a local lawyer) and all that jazz. And then what... Viewer is free, so you have to point to the secondary loss of reputation that affects Manifold's overall reputation, which leads to a battle of expert witnesses in software marketing, which becomes very expensive. Just isn't worth it, which Avast knows perfectly well.
They also know that no matter what chaos they cause, their users won't blame them but will blame the software vendor. For example, your first reaction wasn't "Does Avast not care about telling lies?" but it was "Does Manifold not care about security?" You're not alone in that.
But if you do care about antivirus software not telling lies, it's important that you tell others that whatever Avast does right, it does tell lies by falsely claiming some software packages contain viruses when they don't. "False positive" is just a nice way of saying "it's more profitable for them to lie sometimes than to invest the effort to make sure they're always telling the truth."