I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. The discussion started with cohomology (part 1), continued with bundles and cycles (part 2) and in this part 3, we discuss motivic stuff.

Continue reading «Invariants of projective space III: Motives»

I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. The discussion started with cohomology (part 1) and in this part 2, we discuss intersection-theoretic and bundle-theoretic invariants. In part 3 we will see the motivic stuff.

Continue reading «Invariants of projective space II: Cycles and Bundles»

I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. We will have a look at the cohomology, the Chow groups and the algebraic K-theory of projective space -- a discussion probably interesting to non-motivic people as well. After these invariants, I will look at the motive and the A¹-homotopy type. Then I want to describe the decomposition of the motive (and the homotopy type) homotopy-theoretically, by means of cofiber sequences. (We will see that projective space is not isomorphic to a coproduct of motivic spheres with the same motive). Of course, nothing is new, I'm just working out exercises here.

In this part 1, I discuss only the cohomology of $\mathbb{P}^n$. Part 2 contains a discussion of the intersection theory and bundles and part 3 contains the motivic stuff. I intentionally left out usage of projective bundle formulas, as I will discuss them separately.

Continue reading «Invariants of projective space I: Cohomology»

Two connected compact manifolds N and M are said to be bordant, if there exists a manifold W with boundary consisting of two connected components isomorphic to N and M respectively. The name comes from french and means sharing a boundary. Some people say cobordant, since the manifolds don't share a boundary but "are" shared as a boundary (I don't know how to explain this better than with the definition given above). We will stick to "bordant" because we investigate precisely what "the bordism of a manifold" and "the cobordism of a manifold" are.

One can see that being bordant is an equivalence relation, so it makes sense to speak of bordism classes of manifolds. By enriching N and M with extra structure (like a tangential framing, or an orientation), we get several different notions of bordism classes.

From each of these bordism theories, we get a sequence of spaces $\Omega_n$ such that $\Omega_n$ is the Thom space of a universal bundle over some classifying space (I will explain that later) and $\Sigma \Omega_n$ is homotopy equivalent to $\Omega_{n+1}$. Homotopy theorists like to call such a sequence then a spectrum and by standard theory one can associate to each spectrum a generalized homology theory and a generalized cohomology theory. Even better, Brown's representability theorem states that every generalized (co)homology theory comes from a spectrum, so we have a 1:1 correspondence.

The goal of this article is now to define Thom spectra and to give a geometric interpretation of the corresponding homology and cohomology theories, essentially by carrying out the Pontryagin-Thom construction relatively.

Continue reading «Bordism and Cobordism»