Rational homotopy theory (${\pi }_{\ast }(X)\otimes \mathbb{Q}$)

Study homotopy after killing torsion, especially for simply connected spaces of finite type. The rational Whitehead theorem detects rational homotopy equivalences by rational homology, while Sullivan minimal models encode rational homotopy generators and cohomological relations.

A connected space is nilpotent when its fundamental group is nilpotent and its action on each higher homotopy group is nilpotent: the module has a finite filtration whose successive quotients have trivial group action. Simply connected spaces satisfy these conditions automatically. They are the natural setting for localization and rationalization, since fibrewise coefficient localization is compatible with the successive untwisted stages of such a filtration.

In the simply connected homotopy category, a rational space is a simply connected CW complex whose positive homotopy groups are vector spaces over $\mathbb{Q}$. Multiplication by every nonzero integer on these groups is invertible. More generally one uses nilpotent spaces with localized fundamental group and higher homotopy groups; the simply connected version avoids local-coefficient twisting.

For simply connected $X$, rationalization of a topological space is a map $r:X\to X_{\mathbb{Q}}$ to a rational space inducing ${\pi }_n(X_{\mathbb{Q}})={\pi }_n(X)\otimes \mathbb{Q}$. It is also a rational homology equivalence. Its universal property is the bijection $r^{\ast }:[X_{\mathbb{Q}},Z]\to [X,Z]$ for every rational space $Z$. A Postnikov tower constructs it by replacing each fibre group with its rational tensor product and transporting the Postnikov invariant through the resulting cohomology isomorphism, then taking the homotopy inverse limit.

A map between simply connected spaces is a rational homotopy equivalence when it induces isomorphisms ${\pi }_n(X)\otimes \mathbb{Q}\to {\pi }_n(Y)\otimes \mathbb{Q}$ in every degree. Equivalently it induces an equivalence of their rationalizations, or a rational homology isomorphism by the rational Whitehead theorem. Spaces have the same rational homotopy type when they are connected by a zigzag of such maps. This is distinct from rational homotopy equivalence of algebraic cycles.

The rational homotopy type of a simply connected space is its equivalence class under zigzags of rational homotopy equivalences, or equivalently the homotopy type of its rationalization of a topological space. For the usual finite-type spaces, it is encoded contravariantly by the isomorphism class of the Sullivan minimal model. Isomorphic cohomology rings determine this type only when additional information, such as formality of a topological space, makes the cochain algebra equivalent to the cohomology algebra.

A free graded-commutative rational differential graded algebra with decomposable differentials and a quasi-isomorphism to the rational polynomial forms of a space. For simply connected finite-type spaces, $V^i$ is dual to ${\pi }_i(X)\otimes \mathbb{Q}$. A proposed finite model must have its cohomology and representing quasi-isomorphism checked; matching a list of relations alone is insufficient.

A relative Sullivan algebra is an inclusion $(A,d)\to (A\otimes \Lambda W,D)$ with a well-ordered homogeneous basis of $W$ such that the differential of a generator lies in the algebra on $A$ and earlier generators. It need not be minimal: a differential can have linear terms from earlier generators. Such extensions supply cofibrant replacements for commutative differential graded algebras. A relative acyclic extension of a sphere model represents its path-space fibration and computes a homotopy-fibre model after base change.

For a map $f:E\to B$ with simply connected base and the usual finite-type hypotheses, choose a relative Sullivan algebra $M(B)\to M(B)\otimes \Lambda W$ modelling the path-space fibration, with acyclic total algebra. Tensor it over $M(B)$ with a model of $E$ via $f^{\ast }$. The resulting algebra models the homotopy fibre. Relative Sullivan extensions are cofibrant replacements, so this tensor product computes the derived algebraic pushout, contravariantly corresponding to the topological homotopy pullback. Linear differential pairs must be cancelled before reading rational homotopy groups from generators.

For simply connected rational spaces of finite type, isomorphic minimal Sullivan models are equivalent to the same rational homotopy type. A quasi-isomorphism between minimal models is an isomorphism, so a zigzag of cochain quasi-isomorphisms produces the same minimal model. Existence and uniqueness come from adjoining closed generators to represent missing cohomology and generators with prescribed decomposable differential to kill kernel classes, degree by degree.

For the standard $n$-simplex use ${\Omega }_n=\mathbb{Q}[t_0,\dots ,t_n,d{t}_0,\dots ,d{t}_n]/(\sum t_i-1,\sum d{t}_i)$, with $|t_i|=0$, $|d{t}_i|=1$, and $d(t_i)=d{t}_i$. Compatible families of these forms on the simplices of a simplicial model of $X$ form the commutative differential graded algebra $A_{\mathrm{PL}}(X)$. Its cohomology is $H^{\ast }(X;\mathbb{Q})$: polynomial contraction proves the simplex Poincaré lemma, and extension across faces gives the simplicial de Rham comparison. The contravariant functor takes simplicial pushouts to pullbacks of algebras.

For a degree-one map $f:{\mathbb{CP}}^n\to S^{2n}$ with $n\ge 2$, a minimal model of its homotopy fibre is $(\Lambda (x_2,a_{2n-1},z_{2n+1},b_{4n-2}),d)$ with $dx=dz=0$, $da=x^n$, $db=x^{n-1}z$. Adjoin suspended sphere generators to the projective-space model to model the path pullback, then put $z=y-xa$; the differential becomes the displayed decomposable one. The rational homotopy groups are one-dimensional in those four generator degrees and zero otherwise. At $n=1$ the map is a rational homotopy equivalence and the minimal fibre model is $\mathbb{Q}$.

The quotient ${\mathbb{CP}}^5/{\mathbb{CP}}^2$ is five-connected and has cohomology only in degrees $0,6,8,10$, with positive products zero. Its minimal model therefore starts with closed $a_6,b_8,c_{10}$; no decomposable differential can occur below degree twelve. Sending these three generators to the cohomology basis and all later generators to zero is a chain map and a quasi-isomorphism. Thus its full minimal model is the same as that of $S^6\vee S^8\vee S^{10}$. It starts with generators $u_{11},v_{13},w_{15},t_{15}$ satisfying $du=a^2,dv=ab,dw=ac,dt=b^2$, followed by generators killing remaining products and syzygies.

For a finite wedge sum of spheres of dimensions $n_i\ge 2$, let $L=\mathbb{L}(e_i)$ with $|e_i|=n_i-1$ and zero differential. Its Chevalley–Eilenberg cochains of a graded Lie algebra are a minimal commutative differential graded algebra: one generator dual to each Lie basis word has degree one more than that word, and its differential is dual to the Lie bracket. Its cohomology is $\mathbb{Q}\oplus {⨁}_i\mathbb{Q}{a}_{n_i}$ with all positive products zero. One way to compute this is $U(L)=T(e_i)$; the augmentation module has a length-one free resolution, so the corresponding Ext has only the unit and the dual generating classes.

A space is formal over $\mathbb{Q}$ if its rational cochain commutative differential graded algebra is joined to $(H^{\ast }(X;\mathbb{Q}),0)$ by quasi-isomorphisms. Equivalently in the usual simply connected Sullivan-model setting, its minimal model has a quasi-isomorphism to its cohomology algebra with zero differential. This says its rational homotopy type can be recovered from the cohomology algebra; zero cup products alone do not imply formality of a topological space without an additional argument.

The two quadratic relations form a regular sequence in $\mathbb{Q}[a,b]$, so their Koszul complex has only the quotient-ring cohomology. Represent the two degree-two classes by rational polynomial forms and choose primitives for the exact relations to obtain a quasi-isomorphism. There are no further generators.