Let $X$ be a polyhedron. For each $n$-dimensional face $f$ of $X$ fix a homeomorphism $\sigma_f:\triangle^n\to f$ where $\triangle^n$ is the standard $n-$simplex so that whenever $f$ is a face of $f'$ the map $\sigma_{f'}^{-1}\sigma_f$ is an affine map $\triangle^n\to\triangle^{n'}$ where $n'=\dim \triangle'$. These maps enable one to speak of forms on the simplices of $X$ with rational polynomial coefficients. A degree $n$ Sullivan piecewise linear form on $X$ is a choice of rational polynomial forms, one for each $n$-face of $X$, that agree on the intersections of the faces. Such forms form a commutative differential graded algebra (cdga) denoted $A_{PL}(X)$.

$X$ is called formal if the cdga's $A_{PL}(X)$ and $H^\ast(X,\mathbf{Q})$ (with zero differential) are quasi-isomorphic i.e. can be connected by a chain of cdga quasi-isomorphisms. One topological consequence of formality is that if $X$ is formal, all higher Massey products on $H^\ast(X,\mathbf{Q})$ vanish (see e.g. Griffiths, Morgan, Rational homotopy theory and differential forms). I vaguely remember having heard that the converse is false: there are non-formal spaces with vanishing Massey products. I would like to ask if anyone knows a (preferably not too expensive) example of such a space.

Remark: this question can of course be stated in terms of cdga's without referring to spaces: by Sullivan's realization theorem (Infinitesimal computations in topology, \S 8) to give an example it would suffice to construct a non-formal cdga $A$ with $A^0=\mathbf{Q},A^1=0$ all of whose Massey products vanish. However, if there is a geometric example, I'd be interested to know.

Let $X$ be a polyhedron. For each $n$-dimensional face $f$ of $X$ fix a homeomorphism $\sigma_f:\triangle^n\to f$ where $\triangle^n$ is the standard $n-$simplex so that whenever $f$ is a face of $f'$ the map $\sigma_{f'}^{-1}\sigma_f$ is an affine map $\triangle^n\to\triangle^{n'}$ where $n'=\dim \triangle'$. These maps enable one to speak of forms on the simplices of $X$ with rational polynomial coefficients. A degree $n$ Sullivan piecewise polynomial form on $X$ is a choice of degree $n$ rational polynomial forms, one for each face of $X$, that agree on the intersections of the faces. Such forms form a commutative differential graded algebra (cdga) denoted $A_{PL}(X)$.

$X$ is called formal if the cdga's $A_{PL}(X)$ and $H^\ast(X,\mathbf{Q})$ (with zero differential) are quasi-isomorphic i.e. can be connected by a chain of cdga quasi-isomorphisms. One topological consequence of formality is that if $X$ is formal, all higher Massey products on $H^\ast(X,\mathbf{Q})$ vanish (see e.g. Griffiths, Morgan, Rational homotopy theory and differential forms). I vaguely remember having heard that the converse is false: there are non-formal spaces with vanishing Massey products. I would like to ask if anyone knows a (preferably not too expensive) example of such a space.

Remark: this question can of course be stated in terms of cdga's without referring to spaces: by Sullivan's realization theorem (Infinitesimal computations in topology, \S 8) to give an example it would suffice to construct a non-formal cdga $A$ with $A^0=\mathbf{Q},A^1=0$ all of whose Massey products vanish. However, if there is a geometric example, I'd be interested to know.

Let $X$ be a polyhedron. For each $n$-dimensional face $f$ of $X$ fix a homeomorphism $\sigma_f:\triangle^n\to f$ where $\triangle^n$ is the standard $n-$simplex so that whenever $f$ is a face of $f'$ the map $\sigma_{f'}^{-1}\sigma_f$ is an affine map $\triangle^n\to\triangle^{n'}$ where $n'=\dim \triangle'$. These maps enable one to speak of forms on the simplices of $X$ with rational polynomial coefficients. A degree $n$ Sullivan piecewise linear form on $X$ is a choice of forms with rational polynomial coefficients, one for each $n$-face of $X$, that agree on the intersections of the faces. Such forms form a commutative differential graded algebra (cdga) denoted $A_{PL}(X)$.

$X$ is called formal if the cdga's $A_{PL}(X)$ and $H^\ast(X,\mathbf{Q})$ are quasi-isomorphic i.e. can be connected by a chain of cdga quasi-isomorphisms. One topological consequence of formality is that if $X$ is formal, all higher Massey products on $H^\ast(X,\mathbf{Q})$ vanish (see e.g. Griffiths, Morgan, Rational homotopy theory and differential forms). I vaguely remember having heard that the converse is false: there are non-formal spaces with vanishing Massey products. I would like to ask if anyone knows a (preferably not too expensive) example of such a space.

Remark: this question can of course be stated in terms of cdga's without referring to spaces: by Sullivan's realization theorem (Infinitesimal computations in topology, \S 8) to give an example it would suffice to construct a non-formal cdga $A$ with $A^0=\mathbf{Q},A^1=0$ all of whose Massey products vanish. However, if there is a geometric example, I'd be interested to know.

Let $X$ be a polyhedron. For each $n$-dimensional face $f$ of $X$ fix a homeomorphism $\sigma_f:\triangle^n\to f$ where $\triangle^n$ is the standard $n-$simplex so that whenever $f$ is a face of $f'$ the map $\sigma_{f'}^{-1}\sigma_f$ is an affine map $\triangle^n\to\triangle^{n'}$ where $n'=\dim \triangle'$. These maps enable one to speak of forms on the simplices of $X$ with rational polynomial coefficients. A degree $n$ Sullivan piecewise linear form on $X$ is a choice of rational polynomial forms, one for each $n$-face of $X$, that agree on the intersections of the faces. Such forms form a commutative differential graded algebra (cdga) denoted $A_{PL}(X)$.

$X$ is called formal if the cdga's $A_{PL}(X)$ and $H^\ast(X,\mathbf{Q})$ (with zero differential) are quasi-isomorphic i.e. can be connected by a chain of cdga quasi-isomorphisms. One topological consequence of formality is that if $X$ is formal, all higher Massey products on $H^\ast(X,\mathbf{Q})$ vanish (see e.g. Griffiths, Morgan, Rational homotopy theory and differential forms). I vaguely remember having heard that the converse is false: there are non-formal spaces with vanishing Massey products. I would like to ask if anyone knows a (preferably not too expensive) example of such a space.

Remark: this question can of course be stated in terms of cdga's without referring to spaces: by Sullivan's realization theorem (Infinitesimal computations in topology, \S 8) to give an example it would suffice to construct a non-formal cdga $A$ with $A^0=\mathbf{Q},A^1=0$ all of whose Massey products vanish. However, if there is a geometric example, I'd be interested to know.