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Tom Goodwillie
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Let's talk first about the smooth and simply connected case. As you say, Poincare duality for $X$ yields a spherical fibration, or map $X\to BG$. A lifting $X\to BO$ is necessary but not sufficient for a manifold structure. Such a lifting, or \emph{normal structure}"normal structure", determines an element of the surgery obstruction group ($L$-group). A necessary and sufficent condition (assuming the dimension is not too small) for existence of a manifold structure compatible with the given normal structure is the vanishing of this surgery obstruction.

If we say PL or topological instead of smooth, then the relevant normal structure is a lift to $BPL$ or $BTop$ instead of $BO$.

In the non-simply-connected case there are two important differences. First, the surgery obstruction groups are different. (But they depend only on $\pi_1(X)$ and the orientation character $\pi_1(X)\to \lbrace\pm 1\rbrace$ and the mod $4$ class of the dimension.) Second, in order for any of this to work you need to assume that $X$ satisfies Poincare duality in a strong sense (cap product isomorphisms between cohomology with twisted coefficients and homology with twisted coefficients).

In the following sense it is correct to say that the extra complications in the non-simply-connected case are because of local coefficents: In the simply-connected case Poincare duality (say, in the case when the dimension is a multiple of $4$) produces a quadratic form over $\mathbb Z$, and the $L$-groups are what they are because of some stable classification of such forms, and this is a powerful enough invariant because for simply connected spaces homology has a powerful influence on homotopy theory (sorry for the vagueness). But in the general case one needs twisted homology (with coefficients in $\mathbb Z[\pi_1(X)]$-modules) to get that power, so one needs to think about twisted duality and about quadratic forms over $\mathbb Z[\pi_1(X)]$.

Let's talk first about the smooth and simply connected case. As you say, Poincare duality for $X$ yields a spherical fibration, or map $X\to BG$. A lifting $X\to BO$ is necessary but not sufficient for a manifold structure. Such a lifting, or \emph{normal structure}, determines an element of the surgery obstruction group ($L$-group). A necessary and sufficent condition (assuming the dimension is not too small) for existence of a manifold structure compatible with the given normal structure is the vanishing of this surgery obstruction.

If we say PL or topological instead of smooth, then the relevant normal structure is a lift to $BPL$ or $BTop$ instead of $BO$.

In the non-simply-connected case there are two important differences. First, the surgery obstruction groups are different. (But they depend only on $\pi_1(X)$ and the orientation character $\pi_1(X)\to \lbrace\pm 1\rbrace$ and the mod $4$ class of the dimension.) Second, in order for any of this to work you need to assume that $X$ satisfies Poincare duality in a strong sense (cap product isomorphisms between cohomology with twisted coefficients and homology with twisted coefficients).

In the following sense it is correct to say that the extra complications in the non-simply-connected case are because of local coefficents: In the simply-connected case Poincare duality (say, in the case when the dimension is a multiple of $4$) produces a quadratic form over $\mathbb Z$, and the $L$-groups are what they are because of some stable classification of such forms, and this is a powerful enough invariant because for simply connected spaces homology has a powerful influence on homotopy theory (sorry for the vagueness). But in the general case one needs twisted homology (with coefficients in $\mathbb Z[\pi_1(X)]$-modules) to get that power, so one needs to think about twisted duality and about quadratic forms over $\mathbb Z[\pi_1(X)]$.

Let's talk first about the smooth and simply connected case. As you say, Poincare duality for $X$ yields a spherical fibration, or map $X\to BG$. A lifting $X\to BO$ is necessary but not sufficient for a manifold structure. Such a lifting, or "normal structure", determines an element of the surgery obstruction group ($L$-group). A necessary and sufficent condition (assuming the dimension is not too small) for existence of a manifold structure compatible with the given normal structure is the vanishing of this surgery obstruction.

If we say PL or topological instead of smooth, then the relevant normal structure is a lift to $BPL$ or $BTop$ instead of $BO$.

In the non-simply-connected case there are two important differences. First, the surgery obstruction groups are different. (But they depend only on $\pi_1(X)$ and the orientation character $\pi_1(X)\to \lbrace\pm 1\rbrace$ and the mod $4$ class of the dimension.) Second, in order for any of this to work you need to assume that $X$ satisfies Poincare duality in a strong sense (cap product isomorphisms between cohomology with twisted coefficients and homology with twisted coefficients).

In the following sense it is correct to say that the extra complications in the non-simply-connected case are because of local coefficents: In the simply-connected case Poincare duality (say, in the case when the dimension is a multiple of $4$) produces a quadratic form over $\mathbb Z$, and the $L$-groups are what they are because of some stable classification of such forms, and this is a powerful enough invariant because for simply connected spaces homology has a powerful influence on homotopy theory (sorry for the vagueness). But in the general case one needs twisted homology (with coefficients in $\mathbb Z[\pi_1(X)]$-modules) to get that power, so one needs to think about twisted duality and about quadratic forms over $\mathbb Z[\pi_1(X)]$.

Source Link
Tom Goodwillie
  • 55.9k
  • 7
  • 151
  • 240

Let's talk first about the smooth and simply connected case. As you say, Poincare duality for $X$ yields a spherical fibration, or map $X\to BG$. A lifting $X\to BO$ is necessary but not sufficient for a manifold structure. Such a lifting, or \emph{normal structure}, determines an element of the surgery obstruction group ($L$-group). A necessary and sufficent condition (assuming the dimension is not too small) for existence of a manifold structure compatible with the given normal structure is the vanishing of this surgery obstruction.

If we say PL or topological instead of smooth, then the relevant normal structure is a lift to $BPL$ or $BTop$ instead of $BO$.

In the non-simply-connected case there are two important differences. First, the surgery obstruction groups are different. (But they depend only on $\pi_1(X)$ and the orientation character $\pi_1(X)\to \lbrace\pm 1\rbrace$ and the mod $4$ class of the dimension.) Second, in order for any of this to work you need to assume that $X$ satisfies Poincare duality in a strong sense (cap product isomorphisms between cohomology with twisted coefficients and homology with twisted coefficients).

In the following sense it is correct to say that the extra complications in the non-simply-connected case are because of local coefficents: In the simply-connected case Poincare duality (say, in the case when the dimension is a multiple of $4$) produces a quadratic form over $\mathbb Z$, and the $L$-groups are what they are because of some stable classification of such forms, and this is a powerful enough invariant because for simply connected spaces homology has a powerful influence on homotopy theory (sorry for the vagueness). But in the general case one needs twisted homology (with coefficients in $\mathbb Z[\pi_1(X)]$-modules) to get that power, so one needs to think about twisted duality and about quadratic forms over $\mathbb Z[\pi_1(X)]$.