To see what owns the conflicting files:
pacman -Qo /usr/local/share/man
I suspect pacman did not install that, you must have manually installed rust at some point.
That does not mean though there is no issue with rust 1.34.0. Same settings never before installed it in /usr/local, so a rebuild will be needed to get it back to the normal /usr install directory.
Make sure though to know what you install in /usr/local........and best to never install any manually, always let the package manager track all your files.