Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?
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I learnt of much of their properties when looking at the stuff on microbundles. A very clear source for lots of that is J. Lurie, Spring 2009, Topics in Geometric Topology (18.937) , notes for course 18.937. Look on Jacob's website and you will find the course notes. After that look back at the Haupvermutung book. http://www.maths.ed.ac.uk/~aar/books/haupt.pdf For the open problems I have not that long ago seen a list from the topology atlas: http://at.yorku.ca/list/qa.htm I hope this helps. 

