Template:Brookman&Davis&Khan2007

J. Brookman, J. F. Davis and Q. Khan, Manifolds homotopy equivalent to $P^n\#P^n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}P^n\#P^n$, Math. Ann. 338 (2007), no.4, 947–962. MR2317756 (2009d:57039) Zbl 1129.57029

1 Corrections: Unoriented vs Oriented

There are two typos regarding unoriented vs oriented homeomorphism classes.

• On Line 18 of Page 949, replace $I_n$$I_n$ with $\bar{I}_n$$\bar{I}_n$.
• On Line 18 of Page 949, replace $J_n$$J_n$ with $\bar{J}_n$$\bar{J}_n$.

Thanks to Stanley Chang for catching them.

2 Corrections: Orientation Character

There are four minor typos regarding orientation character. The errors are contained entirely in the second paragraph of Remark 10 and do not affect the rest of the paper.

• On Line -5 of Page 955, the first term of the exact sequence should have trivial orientation character, $L_{n+1}(\Zz[\Zz_2*\Zz_2])$$L_{n+1}(\Zz[\Zz_2*\Zz_2])$.
• On Line -4 of Page 955, the second term of the exact sequence should have non-trivial orientation character, $L_{n+2}(\Zz[\Zz] \to \Zz[\Zz_2^- * \Zz_2^-])$$L_{n+2}(\Zz[\Zz] \to \Zz[\Zz_2^- * \Zz_2^-])$.

Consequently, the unwritten connecting homomorphism becomes $\partial: LN_n(\Zz \to \Zz_2^-*\Zz_2^-) \to L_n(\Zz[\Zz_2*\Zz_2])$$\partial: LN_n(\Zz \to \Zz_2^-*\Zz_2^-) \to L_n(\Zz[\Zz_2*\Zz_2])$, which is used in the first paragraph of the remark.

• On Line 2 of Page 956, replace $\Zz[\Zz_2^-*\Zz_2^-]$$\Zz[\Zz_2^-*\Zz_2^-]$ with $\Zz[\Zz_2*\Zz_2]$$\Zz[\Zz_2*\Zz_2]$.
• On Line 2 of Page 956, replace $\Zz[\Zz] \to \Zz[\Zz_2*\Zz_2]$$\Zz[\Zz] \to \Zz[\Zz_2*\Zz_2]$ with $\Zz[\Zz] \to \Zz[\Zz_2^-*\Zz_2^-]$$\Zz[\Zz] \to \Zz[\Zz_2^-*\Zz_2^-]$.

This notation agrees with the conventions of Wall [Wall1999] and Ranicki [Ranicki1981].

3 References

• [Ranicki1981] A. Ranicki, Exact sequences in the algebraic theory of surgery, Princeton University Press, Princeton, N.J., 1981. MR620795 (82h:57027) Zbl 0471.57012
• [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003