## Geometry & Topology

### The Gromoll filtration, $\mathit{KO}$–characteristic classes and metrics of positive scalar curvature

#### Abstract

Let $X$ be a closed $m$–dimensional spin manifold which admits a metric of positive scalar curvature and let $ℛ+(X)$ be the space of all such metrics. For any $g∈ℛ+(X)$, Hitchin used the $KO$–valued $α$–invariant to define a homomorphism $An−1:πn−1(ℛ+(X),g)→KOm+n$. He then showed that $A0≠0$ if $m=8k$ or $8k+1$ and that $A1≠0$ if $m=8k−1$ or $8k$.

In this paper we use Hitchin’s methods and extend these results by proving that

$A 8 j + 1 − m ≠ 0 a n d π 8 j + 1 − m ( ℛ + ( X ) ) ≠ 0$

whenever $m≥7$ and $8j−m≥0$. The new input are elements with nontrivial $α$–invariant deep down in the Gromoll filtration of the group $Γn+1=π0(Diff(Dn,∂))$. We show that $α(Γ8j−58j+2)≠{0}$ for $j≥1$. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

#### Article information

Source
Geom. Topol., Volume 17, Number 3 (2013), 1773-1789.

Dates
Revised: 16 January 2013
Accepted: 8 April 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732618

Digital Object Identifier
doi:10.2140/gt.2013.17.1773

Mathematical Reviews number (MathSciNet)
MR3073935

Zentralblatt MATH identifier
1285.57015

#### Citation

Crowley, Diarmuid; Schick, Thomas. The Gromoll filtration, $\mathit{KO}$–characteristic classes and metrics of positive scalar curvature. Geom. Topol. 17 (2013), no. 3, 1773--1789. doi:10.2140/gt.2013.17.1773. https://projecteuclid.org/euclid.gt/1513732618

#### References

• J F Adams, On the groups $J(X)$, IV, Topology 5 (1966) 21–71
• P Antonelli, D Burghelea, P J Kahn, Gromoll groups, ${\rm Diff}\, S^n$ and bilinear constructions of exotic spheres, Bull. Amer. Math. Soc. 76 (1970) 772–777
• B Botvinnik, P B Gilkey, The eta invariant and metrics of positive scalar curvature, Math. Ann. 302 (1995) 507–517
• D Burghelea, On the homotopy type of ${\rm diff}(M\sp{n})$ and connected problems, Ann. Inst. Fourier $($Grenoble$)$ 23 (1973) 3–17
• D Burghelea, R Lashof, The homotopy type of the space of diffeomorphisms, I, II, Trans. Amer. Math. Soc. 196 (1974) 1–36; ibid. 196 37–50
• J Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970) 5–173
• E B Curtis, Some nonzero homotopy groups of spheres, Bull. Amer. Math. Soc. 75 (1969) 541–544
• D Gromoll, Differenzierbare Strukturen und Metriken positiver Krümmung auf Sphären, Math. Ann. 164 (1966) 353–371
• B Hanke, T Schick, W Steimle, The space of metrics of positive scalar curvature
• A E Hatcher, A proof of the Smale conjecture, ${\rm Diff}(S\sp{3})\simeq {\rm O}(4)$, Ann. of Math. 117 (1983) 553–607
• M W Hirsch, B Mazur, Smoothings of piecewise linear manifolds, Annals of Mathematics Studies 80, Princeton Univ. Press (1974)
• N Hitchin, Harmonic spinors, Advances in Math. 14 (1974) 1–55
• M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504–537
• R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies 88, Princeton Univ. Press (1977)
• T Lance, Differentiable structures on manifolds, from: “Surveys on surgery theory, Vol. 1”, (S Cappell, A Ranicki, J Rosenberg, editors), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 73–104
• R Lashof, M Rothenberg, Microbundles and smoothing, Topology 3 (1965) 357–388
• H B Lawson Jr, M-L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton Univ. Press (1989)
• J P Levine, Lectures on groups of homotopy spheres, from: “Algebraic and geometric topology”, (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 62–95
• W Lück, A basic introduction to surgery theory, from: “Topology of high-dimensional manifolds, No. 1, 2”, (F T Farrell, L G öttsche, W Lück, editors), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002) 1–224
• I Madsen, R J Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies 92, Princeton Univ. Press (1979)
• J W Milnor, Remarks concerning spin manifolds, from: “Differential and Combinatorial Topology”, Princeton Univ. Press (1965) 55–62
• S P Novikov, Differentiable sphere bundles, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 71–96
• P Piazza, T Schick, Groups with torsion, bordism and rho invariants, Pacific J. Math. 232 (2007) 355–378
• J Rosenberg, $C^*$–algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. (1983) 197–212
• T Schick, A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture, Topology 37 (1998) 1165–1168
• S Smale, Generalized Poincaré's conjecture in dimensions greater than four, Ann. of Math. 74 (1961) 391–406
• S Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992) 511–540
• H Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton Univ. Press (1962)
• M Weiss, Sphères exotiques et l'espace de Whitehead, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986) 885–888
• M Weiss, Pinching and concordance theory, J. Differential Geom. 38 (1993) 387–416