Draft:Formal Spaces (Topology)

In rational homotopy theory, formal spaces are a class of rational spaces for which their rational homotopy type is a formal consequence of their cohomology ring with rational coefficients. In other words, the rational homotopy type is completely determined by the cohomology ring with rational coefficients.

Background

The terminology Quillen and Sullivan in their work on rational homotopy theory. In this context, formallity was introduced as a property of certain classes of differential graded algebras (DGAs). Formal differential algebras were used by Quillen to show that any simply connected commutative differential graded algebra (CDGA) of finite type is the cohomology ring of some topological space. It was also used by Deligne, Griffiths, Morgan, and Sullivan to classify the homotopy types of Kähler manifolds.

Formal CDGAs

Given a CDGA, ${\mathcal {A}}$, over a field $k$ we may view its cohomology ring $H^{*}({\mathcal {A}};k)$ as a CDGA with zero differential. If $f:{\mathcal {A}}\to {\mathcal {B}}$ is a morphism of CDGAs then $f$ induces a CDGA morphism $f^{*}:H^{*}({\mathcal {B}})\to H^{*}({\mathcal {A}})$ on cohomology. Note that, by definition, a CDGA with zero differential is its own cohomology. In particular, a CDGA morphism $f:{\mathcal {A}}\to H^{*}({\mathcal {A}})$ induces a morphism $f^{*}:H^{*}({\mathcal {A}})\to H^{*}({\mathcal {A}})$ on cohomology. Note that the cohomology depends on the field $k$, over which ${\mathcal {A}}$ is defined.

A minimal CDGA, ${\mathcal {M}}$, defined over a field $k$ of characteristic zero, is said to be $k$ formal if there exists a CDGA morphism ${\mathcal {M}}\to H^{*}({\mathcal {M}})$ such that the induced map on cohomology is the identity.

The main property of interest in relation to formal CDGAs is the following. If ${\mathcal {A}}$ is a CDGA such that its minimal model is formal, then the homotopy type of ${\mathcal {A}}$ (i.e., the isomorphism class of its minimal model) is a formal consequence of its cohomology.

Characterizations

It turns out that formality is independent of the base field in the following sense. If ${\mathcal {M}}$ is a CDGA defined over $\mathbb {Q}$ then ${\mathcal {M}}$ is $\mathbb {Q}$ formal if and only if it is $k$ formal for some field $k$ of characteristic zero. In this case, we simply say that ${\mathcal {M}}$ is formal. This result was proved independently by Sullivan, Miller and Neisendorfer, and Halperin and Stasheff

Let ${\mathcal {M}}$ be a minimal CDGA and let $V_{i}$ be the subspaces of $i$-th degree elements of ${\mathcal {M}}$. For each $i$ we may consider the space $C_{i}$ of closed elements of $V_{i}$. The following theorem due to Deligne, Griffiths, Morgan, and Sullivan gives another characterization of formality.

Theorem 1 — ${\mathcal {M}}$ is formal if and only if for each $i$ there is a subspace $N_{i}$ of $V_{i}$ with $V_{i}=N_{i}\oplus C_{i}$ such that, every closed element, $\alpha$, of the ideal generated by $\textstyle \bigoplus _{i}N_{i}$ is exact. The choice of $N_{i}$ is equivalent to a choice of morphism ${\mathcal {M}}\to H^{*}({\mathcal {M}})$ which induces the identity map on cohomology.

Formal Spaces

Let $X$ be a rational space, i.e., $X$ is a simply connected CW complex such that, for each $j$, $\pi _{j}(X)$ is a $\mathbb {Q}$-vector space.

Piecewise-polynomial Differential Forms

For a CW complex, $X$, the singular cochain complex with rational coefficients, $C^{*}(X;\mathbb {Q} )$ naturally forms a DGA over $\mathbb {Q}$ with the multiplication given by the cup product. While the cup product extends to a graded commutative product on singular cohomology, the cup product on singular cochains is not graded commutative in general so $C^{*}(X;\mathbb {Q} )$ may only be a DGA and not a CDGA. Instead, we may consider the complex ${\mathcal {A}}^{*}(X)$ of piecewsise-polynomial differential forms, i.e., differential forms which are given by polynomials on each simplex. This can be thought of as an analog of the de Rham complex for rational spaces. This is sometimes called the piecewise linear or PL de Rham complex and denotes ${\mathcal {A}}_{\operatorname {PL} }^{*}(X)$.

This complex has a natural CDGA structure and the cohomology. There is a natural map $\rho :{\mathcal {A}}^{*}(X)\to C^{*}(X;\mathbb {Q} )$ given by integration of forms, i.e., for $\omega \in {\mathcal {A}}^{*}(X)$ and a simplex $\Delta ^{n}$ we have $$\langle \rho (\omega ),\Delta ^{n}\rangle =\int _{\Delta ^{n}}\omega .$$ For rational spaces, we have a Stokes' theorem which tells us that this is a cochain map. In particular, it induces a map on cohomology.

PL de Rham Theorem —  The map $\rho$ induces a CDGA isomorphism on cohomology.

Thus, the cohomology of ${\mathcal {A}}^{*}(X)$ is canonically isomorphic to the singular cohomology and so we may use ${\mathcal {A}}^{*}(X)$ as our CDGA substitute for $C^{*}(X;\mathbb {Q} )$.

Minimal Models

We define the minimal model of a rational space $X$ to be the minimal model, ${\mathcal {M}}_{X}$, of ${\mathcal {A}}^{*}(X)$. Since ${\mathcal {M}}_{X}$ is minimal, we may write $${\mathcal {M}}_{X}=\textstyle \bigwedge ^{\bullet }V^{*}$$ for some $\mathbb {Q}$-vector space $V$. In particular, we may decompose $V=\textstyle \bigoplus _{j}V_{j}$ where $V_{j}$ is the subspace spanned by the degree $j$ generators of ${\mathcal {M}}_{X}$.

Theorem 2 —  There exists an isomorphism $\pi _{j}(X)\cong V_{j}$ as $\mathbb {Q}$-vector spaces

In particular, ${\mathcal {M}}_{X}$ contains the $\mathbb {Q}$-homotopy type of $X$.

Formal Spaces

We say that a rational space $X$ is formal if its minimal model, ${\mathcal {M}}_{X}$ is a formal CDGA. By definition, this means that that the isomorphism class of ${\mathcal {M}}_{X}$ is a formal consequence of the cohomology of ${\mathcal {A}}^{*}(X)$. By the PL de Rham theorem, this is exactly the singular cohomolgy. Moreover, by Theorem 2, the isomorphism class of ${\mathcal {M}}_{X}$ is exactly the $\mathbb {Q}$-homotopy type of $X$. Thus, if $X$ is a formal space, its $\mathbb {Q}$-homotopy type is a formal consequence of its cohomology ring.

Examples

Here are some examples of formal spaces:

• The $n$-sphere $S^{n}$ are formal for $n\geq 2$.
• Complex projective space $\mathbb {CP} ^{n}$.
• Kähler manifolds.
• Simply connected Lie groups.
• The classifying space $BG$ of a Lie group $G$.
• Compact simply connected symmetric spaces (e.g., $S^{n}$, $\mathbb {CP} ^{n}$, Grassmanians).

See Also

• Rational homotopy theory
• Differential graded algebra