#cobordism

[noosphe-re]

Cobordism N 0 in H with two boundary real circles Apanasov, B.. (1997). Geometry and topology of complex hyperbolic and CR-manifolds. Russian Mathematical Surveys - RUSS MATH SURVEY-ENGL TR. 52. 895-928. 10.1070/RM1997v052n05ABEH002084.

[spectrallysequenced]

It's 5am and I keep having to draw pairs of pants because I have to give a talk tmrw.

Real ones will understand

[isomorphismes]

Homology is meant to count submanifolds, up to cobordism.

Ilya Grigoriev

[thetwomeatmeal]

The Steenrod squares were originally defined basically as power operations using the $E_\infty$ structure of $H\mathbb{F}_2$. Surprisingly if you think about it, they’re stable, and so correspond to elements of $H\mathbb{F}_2^*H\mathbb{F}_2$. Here’s my hot take on why they’re stable: power operations on $MO$ are stable by a basically geometric argument(*), and $H\mathbb{F}_2$ is an $E_\infty$ algebra over $MO$, so as long as all the ordinary cohomology classes we care about lift to $MO$-cohomology classes, we can calculate power operations on $H\mathbb{F}_2$ by using the $MO$-power operations on their lifts. As a matter of fact, all the classes you need to define the squares, namely the fundamental class and the cohomology of $\mathbb{R} P^\infty$, do lift to $MO$-cohomology classes, though I don’t yet have a good explanation why. (The $E_\infty$ map $MO \to H\mathbb{F}_2$ splits as a map of spectra, and in fact, $MO \simeq H\mathbb{F}_2 \wedge X$ for a spectrum $X$, but I don’t know if this makes $MO$ an $E_\infty$ algebra over $H\mathbb{F}_2$.) The same thing should work for mod $p$ cohomology using, I think, $MU \wedge S\mathbb{Z}/p$.

So the idea is that we can rewrite the history of topology and take cobordism theories as more basic than ordinary cohomology theories. There are three conceptual advantages to doing this. First, defining power operations for a cohomology theory $E$ in nonzero degrees is equivalent to choosing $E$-orientations of various representations of symmetric groups, viewed as vector bundles over $B\Sigma_n$. Although it’s ‘obvious’ to people with a few years of algebraic topology that every real vector bundle is $H\mathbb{F}_2$-orientable, it is and should be ‘even more obvious’ to Me the Fourth-Year Grad Student that every real vector bundle is $MO$-orientable. Second, one learns at some point (e.g. it’s in Milnor-Stasheff) that the Stiefel-Whitney classes of a real vector bundle can be defined by the formula: $$w_k = \Phi^{-1}\mathrm{Sq}^k(\Phi(1))$, where $\Phi$ is the Thom isomorphism for the bundle. Well, $MO$ has universal characteristic classes for real vector bundles, so you can use this formula to define stable $MO$ operations, starting from characteristic classes. Third, there’s probably a connection between $MU$ power operations and power operations coming up in chromatic homotopy theory (e.g. the $E_n$ power operations studied by Rezk and Ando) -- some of this might be latent in Quillen’s paper, which I still need to read. There’s likewise a hope of using power operations on other Thom spectra to study the A-hat and Witten genera.

(*)The geometric argument, which I got out of tom Dieck’s ‘Steenrod-Operationen in Kobordism-Theorien’, goes like this. Since power operations are multiplicative, to show stability for $MO$-power operations you just have to calculate them on the fundamental class of $S^1$, i.e. show that the map $$MO^1(S^1) \to MO^*(S^1 \times B\Sigma_2) = MO^*(S^1)[x]$$ sends the fundamental class $\tau$ to $x\tau$. Classes in $MO^*(X)$ are represented by cobordism classes of manifolds with singularities mapping to $X$, and the total external squaring operation is roughly represented by sending $M \to X$ to $(M \times M \times E\Sigma_2)_{\Sigma_2} \to (X \times X \times E\Sigma_2)_{\Sigma_2}$. (Since you’re dealing with infinite-dimensional complexes here, you really should be writing these classes as limits of cobordism classes of finite subcomplexes.) The fundamental class in $MO^1(S^1)$ is represented by the inclusion of a point, $\ast\to S^1$. The total internal squaring operation is given by taking an intersection with the diagonal $X \times B\Sigma_2$. Well, $(\ast \times \ast \times E\Sigma_2)_{\Sigma_2}$ isn’t transverse to this diagonal, but there’s a cobordism to something which has transverse intersection to the diagonal that’s exactly represented by the class $x\tau$. For $MU$, you do the same thing with $S^2$ and make sure you’re writing down a complex cobordism, and so on.