Inertia groups of ( n − 1 ) (n-1) -connected 2 ⁢ n 2n -manifolds

In this paper, we compute the inertia groups of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (n-1) -connected, smooth, closed, oriented <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> 2n -manifolds, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n≥ 3 . As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the 126-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group always vanishes for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≠</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mn>9</m:mn> </m:mrow> </m:mrow> </m:math> n≠ 4,8,9 ; for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≫</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> n 0 , this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>8</m:mn> <m:mo>,</m:mo> <m:mn>9</m:mn> </m:mrow> </m:mrow> </m:math> n=4,8,9 , we apply Kreck’s modified surgery and a special case of Crowley’s 𝑄-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>8</m:mn> </m:mrow> </m:mrow> </m:math> n=4,8 , our results recover unpublished work of Crowley–Nagy and Crowley–Olbermann. In contrast, we show that the homotopy and concordance inertia groups of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (n-1) -connected, smooth, closed, oriented <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> 2n -manifolds with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n≥ 3 always vanish.