Skip to main content
MathOverflow

Return to Answer

added 2 characters in body
Source Link
Lennart Meier
Lennart Meier
  • 12.1k
  • 4
  • 73
  • 107

Yes, this has been done. Laures has defined an f-invariant using elliptic cohomology in The topological q-expansion principle. The explanation in Section 2 of his more recent paper on Toda brackets might be a bit easier to read.

Essentially the idea is the following: Look at the Adams-Novikov spectral sequence. A stable homotopy element $x$ has non-vanishing degree iff it is in exact Adams-Novikov filtration 0, i.e. detected in the 0-line. Suppose the degree vanishes. Then $x$ is in Adams-Novikov filtration (at least) 1. The most naive e-invariant is just the reduction of $x$ into the 1-line of the Adams-Novikov SS (which is an $\mathrm{Ext}^1$-term). To make it a bit easier, we map it into the 1-line of the Adams-Novikov spectral sequence with respect K-theory; this does not loose any information, essentially because K-theory carries a height-1 formal group law. Then we go one step further and map it into $\mathbb{Q}/\mathbb{Z}$. This is the Adams e-invariant.

If the e-invariant of $x$ vanishes, $x$ has Adams-Novikov filtration 2. (This happens automatically if the degree of $x$ is positive and even). We map now $x$ into tothe second line of the Adams-Novikov $E^2$-term for elliptic cohomology -- this does not loose information (essentially) because the formal group of elliptic cohomology has height 2. Laures maps this then further into a much more complicated analogue of $\mathbb{Q}/\mathbb{Z}$, essentially a quotient by the ring of divided congruences (of modular forms) by (certain) integral modular forms.

For concrete calculations with the $f$-invariant, von Bodecker's papers (e.g. as referenced in the Toda bracket paper by Laures) contain a wealth of information.

I guess, there are at least two reasons why the f-invariant is less known than the e-invariant: Firstly, the 1-line of the ANSS has a (very important!) geometric interpretation as the image of $J$. Secondly, the Adams-Novikov spectral sequence was not known at the times Adams invented the e-invariant; in particular, its first line was not computed yet. In contrast, since 1981 (by work of Miller, Ravenel, Wilson and Shimomura) the Adams-Novikov 2-line is completely computed. The f-invariant can therefore not longer help with its computation. On the other hand, the f-invariant can be still useful: Say you want to compute a Toda bracket (or a product) whose result lies in Adams-Novikov filtration 2. Then you can use the f-invariant to identify it with other classes already known. This is what happens in Laures' paper referenced above.

Yes, this has been done. Laures has defined an f-invariant using elliptic cohomology in The topological q-expansion principle. The explanation in Section 2 of his more recent paper on Toda brackets might be a bit easier to read.

Essentially the idea is the following: Look at the Adams-Novikov spectral sequence. A stable homotopy element $x$ has non-vanishing degree iff it is in exact Adams-Novikov filtration 0, i.e. detected in the 0-line. Suppose the degree vanishes. Then $x$ is in Adams-Novikov filtration (at least) 1. The most naive e-invariant is just the reduction of $x$ in the 1-line of the Adams-Novikov SS (which is an $\mathrm{Ext}^1$-term. To make it a bit easier, we map it into the 1-line of the Adams-Novikov spectral sequence with respect K-theory; this does not loose any information, essentially because K-theory carries a height-1 formal group law. Then we go one step further and map it into $\mathbb{Q}/\mathbb{Z}$. This is the Adams e-invariant.

If the e-invariant of $x$ vanishes, $x$ has Adams-Novikov filtration 2. (This happens automatically if the degree of $x$ is positive and even). We map now $x$ into to second line of the Adams-Novikov $E^2$-term for elliptic cohomology -- this does not loose information (essentially) because the formal group of elliptic cohomology has height 2. Laures maps this then further into a much more complicated analogue of $\mathbb{Q}/\mathbb{Z}$, essentially a quotient by the ring of divided congruences (of modular forms) by (certain) integral modular forms.

For concrete calculations with the $f$-invariant, von Bodecker's papers (e.g. as referenced in the Toda bracket paper by Laures) contain a wealth of information.

I guess, there are at least two reasons why the f-invariant is less known than the e-invariant: Firstly, the 1-line of the ANSS has a (very important!) geometric interpretation as the image of $J$. Secondly, the Adams-Novikov spectral sequence was not known at the times Adams invented the e-invariant; in particular, its first line was not computed yet. In contrast, since 1981 (by work of Miller, Ravenel, Wilson and Shimomura) the Adams-Novikov 2-line is completely computed. The f-invariant can therefore not longer help with its computation. On the other hand, the f-invariant can be still useful: Say you want to compute a Toda bracket (or a product) whose result lies in Adams-Novikov filtration 2. Then you can use the f-invariant to identify it with other classes already known. This is what happens in Laures' paper referenced above.

Yes, this has been done. Laures has defined an f-invariant using elliptic cohomology in The topological q-expansion principle. The explanation in Section 2 of his more recent paper on Toda brackets might be a bit easier to read.

Essentially the idea is the following: Look at the Adams-Novikov spectral sequence. A stable homotopy element $x$ has non-vanishing degree iff it is in exact Adams-Novikov filtration 0, i.e. detected in the 0-line. Suppose the degree vanishes. Then $x$ is in Adams-Novikov filtration (at least) 1. The most naive e-invariant is just the reduction of $x$ to the 1-line of the Adams-Novikov SS (which is an $\mathrm{Ext}^1$-term). To make it a bit easier, we map it into the 1-line of the Adams-Novikov spectral sequence with respect K-theory; this does not loose any information, essentially because K-theory carries a height-1 formal group law. Then we go one step further and map it into $\mathbb{Q}/\mathbb{Z}$. This is the Adams e-invariant.

If the e-invariant of $x$ vanishes, $x$ has Adams-Novikov filtration 2. (This happens automatically if the degree of $x$ is positive and even). We map now $x$ into the second line of the Adams-Novikov $E^2$-term for elliptic cohomology -- this does not loose information (essentially) because the formal group of elliptic cohomology has height 2. Laures maps this then further into a much more complicated analogue of $\mathbb{Q}/\mathbb{Z}$, essentially a quotient by the ring of divided congruences (of modular forms) by (certain) integral modular forms.

For concrete calculations with the $f$-invariant, von Bodecker's papers (e.g. as referenced in the Toda bracket paper by Laures) contain a wealth of information.

I guess, there are at least two reasons why the f-invariant is less known than the e-invariant: Firstly, the 1-line of the ANSS has a (very important!) geometric interpretation as the image of $J$. Secondly, the Adams-Novikov spectral sequence was not known at the times Adams invented the e-invariant; in particular, its first line was not computed yet. In contrast, since 1981 (by work of Miller, Ravenel, Wilson and Shimomura) the Adams-Novikov 2-line is completely computed. The f-invariant can therefore not longer help with its computation. On the other hand, the f-invariant can be still useful: Say you want to compute a Toda bracket (or a product) whose result lies in Adams-Novikov filtration 2. Then you can use the f-invariant to identify it with other classes already known. This is what happens in Laures' paper referenced above.

Source Link
Lennart Meier
Lennart Meier
  • 12.1k
  • 4
  • 73
  • 107

Yes, this has been done. Laures has defined an f-invariant using elliptic cohomology in The topological q-expansion principle. The explanation in Section 2 of his more recent paper on Toda brackets might be a bit easier to read.

Essentially the idea is the following: Look at the Adams-Novikov spectral sequence. A stable homotopy element $x$ has non-vanishing degree iff it is in exact Adams-Novikov filtration 0, i.e. detected in the 0-line. Suppose the degree vanishes. Then $x$ is in Adams-Novikov filtration (at least) 1. The most naive e-invariant is just the reduction of $x$ in the 1-line of the Adams-Novikov SS (which is an $\mathrm{Ext}^1$-term. To make it a bit easier, we map it into the 1-line of the Adams-Novikov spectral sequence with respect K-theory; this does not loose any information, essentially because K-theory carries a height-1 formal group law. Then we go one step further and map it into $\mathbb{Q}/\mathbb{Z}$. This is the Adams e-invariant.

If the e-invariant of $x$ vanishes, $x$ has Adams-Novikov filtration 2. (This happens automatically if the degree of $x$ is positive and even). We map now $x$ into to second line of the Adams-Novikov $E^2$-term for elliptic cohomology -- this does not loose information (essentially) because the formal group of elliptic cohomology has height 2. Laures maps this then further into a much more complicated analogue of $\mathbb{Q}/\mathbb{Z}$, essentially a quotient by the ring of divided congruences (of modular forms) by (certain) integral modular forms.

For concrete calculations with the $f$-invariant, von Bodecker's papers (e.g. as referenced in the Toda bracket paper by Laures) contain a wealth of information.

I guess, there are at least two reasons why the f-invariant is less known than the e-invariant: Firstly, the 1-line of the ANSS has a (very important!) geometric interpretation as the image of $J$. Secondly, the Adams-Novikov spectral sequence was not known at the times Adams invented the e-invariant; in particular, its first line was not computed yet. In contrast, since 1981 (by work of Miller, Ravenel, Wilson and Shimomura) the Adams-Novikov 2-line is completely computed. The f-invariant can therefore not longer help with its computation. On the other hand, the f-invariant can be still useful: Say you want to compute a Toda bracket (or a product) whose result lies in Adams-Novikov filtration 2. Then you can use the f-invariant to identify it with other classes already known. This is what happens in Laures' paper referenced above.