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Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?

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You might try poking around Ranicki’s website. I am sure their is some material on the categories above. – Spice the Bird Feb 6 '12 at 16:07
What aspects are you looking for? – Tim Porter Feb 6 '12 at 17:35
Basic introductions, like differences, examples and open problems list. – John B Feb 6 '12 at 18:03
up vote 3 down vote accepted

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.

For the open problems I have not that long ago seen a list from the topology atlas:

I hope this helps.

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