Having read some of the research the whole theory is based around a shawl found around the fourth victim belonging to Aaron Kosminski.
Now It could well be him but this "evidence" does not prove this. All it does prove is that the shawl belonged to Aaron (DNA from semen). Aaron Kosminski who by the way had a history with prostitutes. At best this merely confirms that he was acquainted with Eddowes, a prostitute.