# Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.

• Cobordism Theory for DIFF/Differentiable/smooth manifolds

However, there are Topological Manifolds which are not Differentiable Manifolds.

So my question here for experts is that what do I need to beware and pay attention in order to master a cobordism theory of Topological Manifolds? What are the main differences of the computations of the bordism groups for the given following structures:

Say,

1. Cobordism Theory of TOP/topological manifolds

2. Cobordism Theory for PDIFF/piecewise differentiable manifolds

3. Cobordism Theory for PL/piecewise-linear manifolds

p.s. Are there Spin, Pin$$^+$$, and Pin$$^-$$ versions of these cobordism theories of Topological Manifolds computed in the literature explicitly?

• The Manifold Atlas briefly discusses this question: map.mpim-bonn.mpg.de/…. I'd be very surprised if the spin and pin$^\pm$ PL/TOP bordism groups have been computed, but I'm not certain. Jun 28, 2019 at 22:18
• (1) + (3) See the book by Madsen and Milgram. (2) I believe the wikipedia page is inaccurate, because I do not know a definition of a "piecewise differentiable manifold": the composition of piecewise smooth maps is not necessarily piecewise smooth. (Noted here.) But what confuses me is that if you did read and believe Wikipedia, they explain that "PDIFF manifolds" are the same notion as PL manifolds, which makes it clear that the answer to (2) is the same as the answer to (3).
– mme
Jun 29, 2019 at 0:29
• @Mike Miller, if you know the mistake of Wikipedia page can you edit and point out where/which sentences have mistakes and how to modify them to correct? thank you! Jun 29, 2019 at 2:17
• According to the book by Thurston cited by the Wikipedia article PDIFF, piecewise differentiable maps $\mathbb{R}^n \rightarrow \mathbb{R}^n$ are not closed under composition, which precludes there being a notion of "piecewise differentiable manifold". This is also noted in this answer by Goodwillie: mathoverflow.net/a/27673/61785 Jun 29, 2019 at 12:54
• Rather than a category, it seems that piecewise differentiable functions form a profunctor/distributor $\mathrm{PL}^{\mathrm{op}} \times \mathrm{DIFF} \rightarrow \mathbf{Set}$, in the sense that if $f$ is PL $W \rightarrow X$, $g$ PDIFF $X \rightarrow Y$, and $h$ is smooth $Y \rightarrow Z$ then $h \circ g \circ f$ is PDIFF $W \rightarrow Z$. Jun 29, 2019 at 12:57

My memory may be a bit faulty, but I think I should say something since I know quite a lot about calculations in the Top case. There is a 1966 paper "Cobordism of combinatorial manifolds'' of Williamson that works out the PL case geometrically (as in Thom). As follows from a 1966 paper "Cobordism theories" by Browder, Liulevicius, and Peterson, the problem of computing PL or Top cobordism is essentially cohomological at the prime 2, as is worked out in the cited Annals study of Madsen and Milgram. At odd primes, it follows from Kirby-Siebenmann that the Top and PL cases coincide. The calculation there can be tackled by the Adams spectral sequence. I have an unpublished preprint (with Ligaard, Mann, and Milgram) that goes quite a long way towards this calculation, but certainly without full information. I hope to get a paper out eventually. The calculation relies on my paper ''The homology of $$E_{\infty}$$ ring spaces'' in Springer Volume 533 "The homology of iterated loop spaces'' (with Cohen and Lada) that gives calculations of the homology of $$BTop$$ and related spaces that are the essential starting point. A short paper "The Bockstein and the Adams spectral sequence'' by Milgram and myself compares the two cited spectral sequences quite generally. That comparison in the case of $$MTop$$ is surprisingly helpful. But I should apologize for the decades long delay in getting our calculations published.