It bears noting that comprehensive genetic research has been performed in an attempt to verify the paternity of people alleged to be children of Joseph Smith through plural wives, including descendants of Fannie Alger (such as her firstborn, one Orrison Smith, thought to be the most likely candidate). This genetic research has revealed
no descendant of Joseph Smith through any woman other than his lawful wife, Emma Smith.
http://www.deseretnews.com/article/1...226318,00.html
With that said, no one really knows what transpired between Fannie and JS. Fannie herself reportedly refused to discuss the matter even after JS had died (and she was no longer an active member). Most of the statements suggesting that JS had an illicit affair with Fannie, however, were made by members and former members who were disaffected at the time of their statements (e.g., Oliver Cowdery). On the other hand, it appears to be undisputed that Fannie came to the Smith home to work as a young housekeeper and Emma eventually booted her from their home because of Emma's personal dislike for Fannie. Seems strange indeed.
BTW, one of Crompton's works about Fannie Alger can be found here:
http://content.lib.utah.edu/cdm4/doc...CISOSHOW=18163