##### Department of Mathematics,

University of California San Diego

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### Math 216 - Topology learning Seminar

## Henning Hohnhold

#### UCSD

## Universal deformations in algebraic topology: the Hopkins-Miller theorem

##### Abstract:

I'm going to explain the theorem of Hopkins and Miller (and partly Goerss) that gives a version of Lubin-Tate deformation theory in the context of algebraic topology. More concretely, the theorem says that there is a functor $(k,\Gamma) \mapsto E_{(k,\Gamma)}$ from formal groups laws over perfect fields of characteristic $p>0$ to a very nice category of commutative ring spectra, namely $E_{\infty}$-ring spectra. It has the property that the formal group law of the cohomology theory associated with the ring spectrum $E_{(k,\Gamma)}$ is the universal deformation of $(k,\Gamma)$. By functoriality, we obtain an action of the (extended) Morava stabilizer group on the spectrum $E_{(\mathbb{F}_{p^n},H_n)}$, where $H_n$ denotes the Honda formal group law of height $n$. One application is the construction of the higher real $K$-theories $EO_n$ as the homotopy fixed point spectra obtained from the action of finite index subgroups of the Morava stabilizer subgroup.

### March 6, 2007

### 9:30 AM

### AP&M 7218

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