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I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 homotopy theory, and homotopy groups contain more information than homology groups. Is there a relationship between motives and homotopies?

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An MFO report: – Thomas Riepe Feb 14 '11 at 8:55
You say there's only one homotopy theory... but there are tons!! – Fernando Muro Feb 14 '11 at 9:58
You said you've read that "motives were designed to be the common part of the many homology theories", but I think that's not quite correct: motives are designed to contain the (co)homological information of algebraic objects (schemes, varieties). I guess there are a lot of "ill-behaved" topological spaces which don't have the homotopy type of a CW-complex and don't have anything to do with algebraic geometry, where motivic ideas don't apply. On the other hand, if anyone knows of a theory of motives for arbitrary topological spaces, I'd be interested :-) – Konrad Voelkel Feb 14 '11 at 10:16
@Aaron - I disagree that a spectrum is "the" homotopy theoretic generalisation of a topological space. It is only a stable homotopy theoretic generalisation of a topological space. Or rather, that a topological space gives rise to a rather special example of a spectrum. – David Roberts Feb 17 '11 at 22:40
So you are asking if their is a theory of motives of topological spaces? or CW complexes? or spectra? I think that is a very different issue then what motivic homotopy theory deals with. Motivic homotopy theory is about homotopy theory of schemes. It also does not provide a category of (mixed) motives that Weil was interested in having produced. – Sean Tilson Feb 18 '11 at 7:07

In algebraic topology, there is a close relationship between stable homotopy theory and the study of (generalized) cohomology theories. Basically, all the cohomology theories become representable on the stable category of spectra and so, from the point of view of stable homotopy theory, the study of cohomology theories can be viewed as the study of their representing spectra.

More recently it has been discovered (through the work of Voevodsky and others) that there is an analogous situation in algebraic geometry. Keywords to look up: motivic homotopy theory, $\mathbb{A}^1$-homotopy theory. Basically, we can construct a homotopy theory for algebraic varieties and a suitable homotopy category which plays a role analogous to the stable category of spectra in topology. One thing this gives us is that it enables us to define new cohomology theories for algebraic varieties by describing their representing spectra ("motivic spectra"). For example, motivic cohomology, algebraic K-theory, and algebraic cobordism can be constructed in this way. This whole circle of ideas is closely related to recent work on motives and motivic cohomology. For example, Voevodsky's construction of a "derived category of mixed motives" is closely related to this work.

The following is a very easy-going introduction to the idea of motivic homotopy theory and is understandable even by an undergraduate:

  • Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002 by Bjørn Dundas, Marc Levine, Paul Østvær, Oliver Röndigs and Vladimir Voevodsky

It is also worth reading Voevodsky's 1998 ICM address:

  • Vladimir Voevodsky - A^1-homotopy theory (Proceedings of the 1998 ICM)

There is a lot more that could be said about this very interesting area of mathematics.

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