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finally an Idea-section at Poincaré duality.
(Needs more work, clearly, but should be a start)
finally added Definition and statement, including the full chain-level statment.
Fair enough. So I have changed it to the following now:
Around 1895 Henri Poincaré made an observation about Betti numbers of closed manifolds, which in the 1930 was the formulated by Eduard Čech and Hassler Whitney in the following modern form:
Jim, you are undoubtedly right, but this raises the question: what role did Poincaré play in this (or why was this result named in his honor)? Perhaps he enunciated a kind of shadow of this result involving Betti numbers, or something like that?
Edit: Urs beat me to it by seconds.
Poincaré was trying to relate homology in degrees $k$ and $n - k$, essentially via the idea of reciprocal polyhedra. He got in rather a tangle with torsion. It’s all at around p. 28 of Dieudonné’s A History of Algebraic and Differential Topology, 1900 - 1960.
Thanks, David, I have added that pointer to the References-section.
Hi Urs, can you tell me again what you said yesterday after Colin McLarty’s talk about this. So you said there’s a hyperdoctrine of spectra over spaces, and that left and right adjoint to base change $X \to 1$, gives homology and cohomology. And then for ’self-dual’ $X$ these coincide, otherwise known as orientation. Something like that?
I can’t remember how you said manifolds come out of this. Was it that among spaces which are manifolds, the orientable ones are dual?
Would that be worth spelling out? I’m sure you say it elsewhere, but it gets a little lost among talk of $\chi$-twisted such-and-such.
What happens then if you choose to base change along a general $X \to Y$?
Ah, of course there’s something on one aspect above: Poincaré duality space gives a name to that kind of space, and Poincaré complex tells us that one which is a CW-complex behaves like a closed manifold.
Since I came across nonabelian Poincaré duality just now, is there a word to say why it resembles ordinary Poincaré duality?
The modern picture of Poincaré-duality in generalized cohomology is around corollary 4.1.16 in master thesis Nuiten (schreiber) (see the remark on the top of the next page). The neat general abstract picture underlying this is the content of Quantization via Linear homotopy types (schreiber).
A table surveying the relation between concepts in dependent linear type theory and twisted generalized cohomology theory is at twisted generalized cohomology in linear homotopy type theory – table.
When I find a second I’ll write the statement in question out clearly somewhere on the $n$Lab. Not today, probably.
Ok, whenever you have time.
Looking again at some of those links, I can see there’s lots I haven’t taken in. Any clue why it’s the cohomology side (“dual of dependent sum”, so also “dependent product of dual”) that gets the richer ring structure?
Thanks. (These misprints are not due to typos but a software glitch, which at one point in the past replaced non-ascii special characters by question marks.)
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