Return to Answer

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I had forgotten that I had a manifold interpretation of the vanishing of the rational total surgery obstruction in Proposition 7.7.5 (page 763) of my book on exact sequences in the algebraic theory of surgery (reference 2 below)! So I have now edited my original answer accordingly

The total surgery obstruction $s^R(X) \in {\mathbb S}_n(X;R)$ of an $R$-coefficient Poincare duality space $X$ can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples). Proposition 7.7.5 of 2 does give a manifold interpretation of $s^R(X)=0 \in {\mathbb S}_n(X;R)$, but I am not sure if it is sufficiently geometric for practical purposes.

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

[1 "The total surgery obstruction"]1 1978 paper

[2 "Exact sequences in the algebraic theory of surgery"]2 1982 book

[3 "Algebraic L-theory and topological manifolds"]3 1992 book

[4 "The total surgery obstruction"]4 2010 MPIM lecture

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I had forgotten that I had a manifold interpretation of the vanishing of the rational total surgery obstruction in Proposition 7.7.5 (page 763) of my 1980 book on exact sequences in the algebraic theory of surgery (reference 2 below)! So I have now edited my original answer accordingly

The total surgery obstruction $s^R(X) \in {\mathbb S}_n(X;R)$ of an $R$-coefficient Poincare duality space $X$ can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples). Proposition 7.7.5 of 2 does give a manifold interpretation of $s^R(X)=0 \in {\mathbb S}_n(X;R)$, but I am not sure if it is sufficiently geometric for practical purposes.

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

[1 "The total surgery obstruction"]1 1978 paper

[2 "Exact sequences in the algebraic theory of surgery"]2 1980 book

[3 "Algebraic L-theory and topological manifolds"]3 1992 book

[4 "The total surgery obstruction"]4 2010 MPIM lecture

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I do not know a manifold interpretation of the vanishing of the rational total surgery obstruction. However, the total surgery obstruction of an $R$-coefficient Poincare duality space can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples).

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

The total surgery obstruction (1978) paper

Algebraic L-theory and topological manifolds (1992) book

The total surgery obstruction (2010) slides of MPIM lecture

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I had forgotten that I had a manifold interpretation of the vanishing of the rational total surgery obstruction in Proposition 7.7.5 (page 763) of my book on exact sequences in the algebraic theory of surgery (reference 2 below)! So I have now edited my original answer accordingly

The total surgery obstruction $s^R(X) \in {\mathbb S}_n(X;R)$ of an $R$-coefficient Poincare duality space $X$ can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples). Proposition 7.7.5 of 2 does give a manifold interpretation of $s^R(X)=0 \in {\mathbb S}_n(X;R)$, but I am not sure if it is sufficiently geometric for practical purposes.

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

[1 "The total surgery obstruction"]1 1978 paper

[2 "Exact sequences in the algebraic theory of surgery"]2 1982 book

[3 "Algebraic L-theory and topological manifolds"]3 1992 book

[4 "The total surgery obstruction"]4 2010 MPIM lecture

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I do not know a manifold interpretation of the vanishing of the rational total surgery obstruction. However, the total surgery obstruction of an $R$-coefficient Poincare duality space can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples).

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I do not know a manifold interpretation of the vanishing of the rational total surgery obstruction. However, the total surgery obstruction of an $R$-coefficient Poincare duality space can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples).

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

The total surgery obstruction (1978) paper

Algebraic L-theory and topological manifolds (1992) book

The total surgery obstruction (2010) slides of MPIM lecture