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The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products. This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.

The rational cohomology algebra of that manifold is generated by symbols $\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by $$ \nu_{ijk}\nu_{ijl}=0. $$ See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero..... wait, I need to check this.

I forget if there is a good argument about why the Massey products should vanish. But I redid the computation $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$ and found zero. The way I did the computation was by representing the cohomology classes by explicit submanifolds, and doing intersections.

More precisely:
• $\nu_{123}$ is represented by the submanifold $M_{123}$ where the marked points "0" and "1" are separated from the marked points "2" and "3" by a normal crossing.
• $\nu_{234}$ is represented by the submanifold $M_{234}$ where the marked points "0" and "2" are separated from the marked points "3" and "4" by a normal crossing.
• $\nu_{345}$ is represented by the submanifold $M_{345}$ where the marked points "0" and "3" are separated from the marked points "4" and "5" by a normal crossing.
At this point, one needs to check that the manifolds $M_{123}$ and $M_{234}$ are transverse, and that the manifolds $M_{234}$ and $M_{345}$ are transverse (a necessary condition for this kind of computation to work out).

• The intersection $M_{123}\cap M_{234}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "1", "2" are separated from the points "3", "4". Call that manifold $A$.
• The intersection $M_{234}\cap M_{345}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "2" are separated from the points "3", "4", "5". Call that manifold $B$.

Finally, we need to compute the cohomology class of $X := (M_{123}\cap B) \cup (A \cap M_{234})$. That manifold represents the Massey product $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$. Unfortunately, $X$ is not empty. But you can intersect it with all the generators of $H_2$ and see that you always get zero. The generators of H_2 are the tori I talked about in the beginning of my answer. The manifold $X$ is actually disjoint from all but one of these tor, and intersects that last torus non-transversely. Even worse: it is contained in that torus. One then needs to deform it and see that it can be made disjoint from that last torus.

I guess I should also check product $\langle \nu_{123}, \nu_{234}, \nu_{123}\rangle$...

The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products. This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.

The rational cohomology algebra of that manifold is generated by symbols $\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by $$ \nu_{ijk}\nu_{ijl}=0. $$ See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero..... wait, I need to check this.

I forget if there is a good argument about why the Massey products should vanish. But I redid the computation $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$ and found zero. The way I did the computation was by representing the cohomology classes by explicit submanifolds, and doing intersections.

More precisely:
• $\nu_{123}$ is represented by the submanifold $M_{123}$ where the marked points "0" and "1" are separated from the marked points "2" and "3" by a normal crossing.
• $\nu_{234}$ is represented by the submanifold $M_{234}$ where the marked points "0" and "2" are separated from the marked points "3" and "4" by a normal crossing.
• $\nu_{345}$ is represented by the submanifold $M_{345}$ where the marked points "0" and "3" are separated from the marked points "4" and "5" by a normal crossing.
At this point, one needs to check that the manifolds $M_{123}$ and $M_{234}$ are transverse, and that the manifolds $M_{234}$ and $M_{345}$ are transverse (a necessary condition for this kind of computation to work out).

• The intersection $M_{123}\cap M_{234}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "1", "2" are separated from the points "3", "4". Call that manifold $A$.
• The intersection $M_{234}\cap M_{345}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "2" are separated from the points "3", "4", "5". Call that manifold $B$.

Finally, we need to compute the cohomology class of $X := (M_{123}\cap B) \cup (A \cap M_{234})$. That manifold represents the Massey product $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$. Unfortunately, $X$ is not empty. But you can intersect it with all the generators of $H_2$ and see that you always get zero. The generators of H_2 are the tori I talked about in the beginning of my answer. The manifold $X$ is actually disjoint from all but one of these tor, and intersects that last torus non-transversely. Even worse: it is contained in that torus. One then needs to deform it and see that it can be made disjoint from that last torus.

I guess I should also check the Massey product $\langle \nu_{123}, \nu_{234}, \nu_{123}\rangle$...

The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products. This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.

The rational cohomology algebra of that manifold is generated by symbols $\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by $$ \nu_{ijk}\nu_{ijl}=0. $$ See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero..... wait, I need to check this.

I forget if there is a good argument about why the Massey products should vanish. But I redid the computation $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$ and found zero. The way I did the computation was by representing the cohomology classes by explicit submanifolds, and doing intersections.

More precisely:
• $\nu_{123}$ is represented by the submanifold $M_{123}$ where the marked points "0" and "1" are separated from the marked points "2" and "3" by a normal crossing.
• $\nu_{234}$ is represented by the submanifold $M_{234}$ where the marked points "0" and "2" are separated from the marked points "3" and "4" by a normal crossing.
• $\nu_{345}$ is represented by the submanifold $M_{345}$ where the marked points "0" and "3" are separated from the marked points "4" and "5" by a normal crossing.
At this point, one needs to check that the manifolds $M_{123}$ and $M_{234}$ are transverse, and that the manifolds $M_{234}$ and $M_{345}$ are transverse (a necessary condition for this kind of computation to work out).

• The intersection $M_{123}\cap M_{234}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "1", "2" are separated from the points "3", "4". Call that manifold $A$.
• The intersection $M_{234}\cap M_{345}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "2" are separated from the points "3", "4", "5". Call that manifold $B$.

Finally, we need to compute the cohomology class of $X := (M_{123}\cap B) \cup (A \cap M_{234})$. That manifold represents the Massey product $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$. Unfortunately, $X$ is not empty. But you can intersect it with all the generators of $H_2$ and see that you always get zero. The denerators of H_2 are the tori I talked about in the beginning of my answer. The manifold $X$ is actually disjoint from all but one of these tor, and intersects that last torus non-transversely. Even worse: it is contained in that torus. One then needs to deform it and see that it can be made disjoint from that last torus.

The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products. This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.

The rational cohomology algebra of that manifold is generated by symbols $\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by $$ \nu_{ijk}\nu_{ijl}=0. $$ See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero..... wait, I need to check this.

I forget if there is a good argument about why the Massey products should vanish. But I redid the computation $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$ and found zero. The way I did the computation was by representing the cohomology classes by explicit submanifolds, and doing intersections.

More precisely:
• $\nu_{123}$ is represented by the submanifold $M_{123}$ where the marked points "0" and "1" are separated from the marked points "2" and "3" by a normal crossing.
• $\nu_{234}$ is represented by the submanifold $M_{234}$ where the marked points "0" and "2" are separated from the marked points "3" and "4" by a normal crossing.
• $\nu_{345}$ is represented by the submanifold $M_{345}$ where the marked points "0" and "3" are separated from the marked points "4" and "5" by a normal crossing.
At this point, one needs to check that the manifolds $M_{123}$ and $M_{234}$ are transverse, and that the manifolds $M_{234}$ and $M_{345}$ are transverse (a necessary condition for this kind of computation to work out).

• The intersection $M_{123}\cap M_{234}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "1", "2" are separated from the points "3", "4". Call that manifold $A$.
• The intersection $M_{234}\cap M_{345}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "2" are separated from the points "3", "4", "5". Call that manifold $B$.

Finally, we need to compute the cohomology class of $X := (M_{123}\cap B) \cup (A \cap M_{234})$. That manifold represents the Massey product $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$. Unfortunately, $X$ is not empty. But you can intersect it with all the generators of $H_2$ and see that you always get zero. The generators of H_2 are the tori I talked about in the beginning of my answer. The manifold $X$ is actually disjoint from all but one of these tor, and intersects that last torus non-transversely. Even worse: it is contained in that torus. One then needs to deform it and see that it can be made disjoint from that last torus.

I guess I should also check product $\langle \nu_{123}, \nu_{234}, \nu_{123}\rangle$...

The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products. This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.

The rational cohomology algebra of that manifold is generated by symbols $\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by $$ \nu_{ijk}\nu_{ijl}=0. $$ See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero..... wait, I need to check this.

The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products. This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.

The rational cohomology algebra of that manifold is generated by symbols $\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by $$ \nu_{ijk}\nu_{ijl}=0. $$ See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero..... wait, I need to check this.

I forget if there is a good argument about why the Massey products should vanish. But I redid the computation $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$ and found zero. The way I did the computation was by representing the cohomology classes by explicit submanifolds, and doing intersections.

More precisely:
• $\nu_{123}$ is represented by the submanifold $M_{123}$ where the marked points "0" and "1" are separated from the marked points "2" and "3" by a normal crossing.
• $\nu_{234}$ is represented by the submanifold $M_{234}$ where the marked points "0" and "2" are separated from the marked points "3" and "4" by a normal crossing.
• $\nu_{345}$ is represented by the submanifold $M_{345}$ where the marked points "0" and "3" are separated from the marked points "4" and "5" by a normal crossing.
At this point, one needs to check that the manifolds $M_{123}$ and $M_{234}$ are transverse, and that the manifolds $M_{234}$ and $M_{345}$ are transverse (a necessary condition for this kind of computation to work out).

• The intersection $M_{123}\cap M_{234}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "1", "2" are separated from the points "3", "4". Call that manifold $A$.
• The intersection $M_{234}\cap M_{345}$ is bounded by $\frac {1} {2}$ times the manifold where the points "0", "2" are separated from the points "3", "4", "5". Call that manifold $B$.

Finally, we need to compute the cohomology class of $X := (M_{123}\cap B) \cup (A \cap M_{234})$. That manifold represents the Massey product $\langle \nu_{123}, \nu_{234}, \nu_{345}\rangle$. Unfortunately, $X$ is not empty. But you can intersect it with all the generators of $H_2$ and see that you always get zero. The denerators of H_2 are the tori I talked about in the beginning of my answer. The manifold $X$ is actually disjoint from all but one of these tor, and intersects that last torus non-transversely. Even worse: it is contained in that torus. One then needs to deform it and see that it can be made disjoint from that last torus.