Mini Course: Infinite loop spaces and positive scalar curvature - By Johannes Ebert (University of Münster)

The first lecture will take place at the Topology seminar Monday at 15:15. The other lectures will be from 11:00-12:00 on Tuesday and Wednesday in aud 5 and on Thursday and Friday aud 6.

The purpose of this lecture series is to give an overview of my recent work with Botvinnik and Randal-Williams (arXiv:1411.7408)

on spaces of positive scalar curvature metrics on high-dimensional spin manifolds.
For a closed $d$-dimensional spin manifold $W$, there is a secondary index invariant $\mathrm{inddiff}: \mathcal{R}^+ (W^d) \to \Omega^{\infty+d+1} \mathbf{KO}$, from the space of metrics of positive scalar curvature metrics on $W$ to a suitable space of the real $K$-Theory spectrum.
The homotopy groups of $\Omega^{\infty+d+1} \mathbf{KO}$ are well-known due to Bott periodicity, and we ask which homotopy classes are hit by $\mathrm{inddiff}$.

Our main result is that for $d = 2n \geq 6$, the natural orientation map $\Omega \hat{\mathscr{A}} : \Omega^{\infty+1} \mathrm{MTSpin} (2n) \to \Omega^{\infty+2n+1} \mathbf{KO}$ factors (up to homotopy) through the space $\mathcal{R}^+ (W^{2n})$, and a similar result holds for odd-dimensional manifolds (of dimension $\geq 7$).
Using traditional methods from algebraic topology, one can derive detection theorems for $\mathrm{inddiff}$ (which supersede the previously known results).

I will try to give a overview of the techniques used in the proof, which rests on three pillars.
The first is index theory for Dirac operators (which is relevant due to the appearance of the scalar curvature as the remainder term in the Weitzenb\"ock formula for the spin Dirac operator).
Besides the Atiyah-Singer family index theorem, the additivity theorem for the index and the spectral flow theorem play a key role.

The second pillar is the Gromov-Lawson surgery method for metrics of positive scalar curvature. The original statement was that if a manifold $W$ admits a metric of positive scalar curvature and is altered by a surgery in codimension $\geq 3$, then the resulting manifold $W'$ also does have a metrics of positive scalar curvature. This result has been improved considerably by Chernysh and Walsh and in fact the homotopy type of the space $\mathcal{R}^+ (W)$ only depends on the spin cobordism class of $W$ if $W$ is highdimensional and simply connected.

The third pillar is the high-dimensional analogues of the Madsen-Weiss theorem and the Harer stability theorem, which both were proven by Galatius and Randal-Williams.
Their results show that the space $\Omega^{\infty} \mathrm{MTSpin(2n)}$ can be homologically approximated by classifying spaces of diffeomorphism groups of certain $2n$-dimensional spin manifolds.

All these different ingredients are then integrated by methods from homotopy theory.