Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 24 (Rational Homotopy Theory)

Work in the homotopy category of simply connected CW complexes. This is one of the permitted assumptions on the fundamental group; it makes every coefficient system below untwisted. A rational space is a simply connected space whose homotopy groups are vector spaces over $\mathbb{Q}$. A rationalization of a topological space is a map $r:X\to X_{\mathbb{Q}}$ to such a space which inducesIts purpose includes a universal mapping property, which will be described after the construction.

An Eilenberg–MacLane space $K(A,n)$ has exactly one nonzero homotopy group, ${\pi }_n=A$, where $A$ is abelian when $n\ge 2$. The representability theorem says $[B,K(A,n)]\cong H^n(B;A)$, naturally in $B$ and $A$. More generally, a principal fibration is a homotopy pullback of a path-space fibration $PC\to C$ along a map from its base to $C$; the fibre $\Omega C$ acts on the fibre paths by concatenation. For the Postnikov tower construction take $C=K(A,n+1)$. A principal Eilenberg–MacLane fibration with fibre $K(A,n)$ is the homotopy pullback of the path fibration of $K(A,n+1)$. Equivalently, its total space is the homotopy fibre of a map $k:B\to K(A,n+1)$. It is classified by the cohomology class $[k]\in H^{n+1}(B;A)$; there is no nontrivial action of the base fundamental group on the fibre coefficients in this principal description.

The Postnikov tower theorem supplies spaces $P_{n}X$ and compatible mapswith ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$ and zero for $i>n$. For a simply connected $X$, the fibre of $P_{n}X\to P_{n-1}X$ is $K({\pi }_{n}X,n)$, and the fibration is principal. Its Postnikov invariant isrepresented by $P_{n-1}X\to K({\pi }_{n}X,n+1)$, whose homotopy fibre is $P_{n}X$. Thus the homotopy groups alone are not the tower data: the classes $k_n$ specify how each new group is attached. For instance a zero class gives a product with $K({\pi }_{n}X,n)$.

State also the standard rational comparison results used in constructing the tower: $K(A,n)\to K(A\otimes \mathbb{Q},n)$ is a rational homology equivalence; a map between simply connected spaces is a rational homology equivalence if and only if it induces isomorphisms on all rational homotopy groups; and a rational homology equivalence induces isomorphisms in cohomology with any constant rational vector-space coefficients. The last assertion uses the universal coefficient theorem over the field $\mathbb{Q}$. These results are used here without proof, as allowed for this essay.

Construct rational stages inductively, with $P_1^{\mathbb{Q}}=\ast$. Suppose $r_{n-1}:P_{n-1}X\to P_{n-1}^{\mathbb{Q}}$ has already been constructed and is a rational homology equivalence. Put $V_n={\pi }_{n}X\otimes \mathbb{Q}$ and change coefficients in $k_n$ to obtain ${\overline{k}}_n\in H^{n+1}(P_{n-1}X;V_n)$. The comparison result gives an isomorphismLet $k_n^{\mathbb{Q}}$ be its unique inverse image and defineEquality of the pulled-back classes gives a homotopy-commuting square of classifying maps, and hence a map $r_n:P_{n}X\to P_n^{\mathbb{Q}}$ over $r_{n-1}$. On the fibre it is the map induced by ${\pi }_{n}X\to V_n$. The homotopy exact sequences therefore show that $r_n$ tensors all homotopy groups through degree $n$ with $\mathbb{Q}$, while the higher groups of both stages vanish. The rational comparison theorem makes $r_n$ a rational homology equivalence, allowing the induction to continue.

Take the homotopy inverse limit of the rational tower:The convergence theorem for a Postnikov tower gives $X\simeq {\mathrm{holim}}_n{P}_{n}X$ and the compatible stage maps induce $r:X\to X_{\mathbb{Q}}$. To see what convergence says here, fix a homotopy degree $i$. From stage $n\ge i$ onwards the group and its transition maps are constant; the inverse-limit group is ${\pi }_{i}X\otimes \mathbb{Q}$ and the associated derived-limit obstruction is zero. Thus this construction has exactly the claimed rational homotopy groups. Taking a homotopy limit, rather than an arbitrary set-theoretic inverse limit, is part of the construction.

The rational localization theorem says that this map has the universal propertyIn other words every map from $X$ to a rational space factors through $r$, and the factorization is unique up to homotopy. This concerns homotopy classes, not strict uniqueness of point-set maps. The theorem can be formulated as localization of the simply connected homotopy category at rational homotopy equivalences; rational spaces are its local objects. It also implies uniqueness of rationalization of a topological space: apply the property to two candidate rational spaces to obtain inverse equivalences under $X$. In particular the construction forgets torsion in homotopy while retaining precisely the information needed for maps to rational targets.

The same theory extends to nilpotent spaces, where the fundamental group is nilpotent and its action on higher homotopy groups is nilpotent. That version requires localization of the fundamental group and compatible treatment of its coefficient action. The simply connected assumption chosen here avoids those additional issues and supplies the full construction and mapping property requested.

For based maps $\alpha :S^p\to Y$ and $\beta :S^q\to Y$, use the standard cell decompositionThe attaching map $w:S^{p+q-1}\to S^p\vee S^q$ is obtained from the boundary of $D^p\times D^q$, collapsing the appropriate coordinate boundaries. The geometric Whitehead product is the homotopy classOrient the product cell consistently with the ordinary product orientation. Its class depends only on the two homotopy classes, and it is natural: $f_{\ast }[\alpha ,\beta {]}_W=[f_{\ast }\alpha ,f_{\ast }\beta {]}_W$.

For $p,q\ge 2$ it is additive in both variables:Thus it is bilinear over $\mathbb{Z}$, and after rationalization of a topological space over $\mathbb{Q}$. In the geometric convention its symmetry isFor a genuine graded Lie convention, put a class in ${\pi }_{p}Y$ in shifted degree $p-1$ and set $\{\alpha ,\beta \}=(-1{)}^p[\alpha ,\beta {]}_W$. Equivalently transport the Samelson product through reduced suspension and loop space adjunction. Writing $|\alpha |=p-1$, this bracket has degree zero, satisfiesand has the graded Jacobi identityThese are two equivalent sign presentations of the same Whitehead product. In particular the square of an odd-dimensional geometric class is two-torsion, whereas an even-dimensional sphere class can have a nonzero rational square. Reduced suspension kills Whitehead products, since the suspended attaching map of the product cell is null-homotopic. No proofs of these remaining algebraic properties are needed here.

Bilinearity has a specific fundamental-group exception. If both inputs lie in ${\pi }_{1}Y$, the attaching loop of the torus is the commutator, so the operation is $gh{g}^{-1}{h}^{-1}$, with the possible inverse corresponding to reversal of its boundary orientation. This is not a homomorphism in either input in a general nonabelian group.

If one input is $g\in {\pi }_{1}Y$ and the other is $\beta \in {\pi }_{q}Y$, $q\ge 2$, the product records the action difference $g\cdot \beta -\beta$, up to the fixed overall boundary sign. Choose the action convention matching that boundary orientation and write ${\delta }_g\beta =g\cdot \beta -\beta$. ThenThe second formula is a crossed-additivity law, not ordinary additivity in $g$. It follows by expanding $gh\cdot \beta -\beta =g\cdot (h\cdot \beta -\beta )+(g\cdot \beta -\beta )$. Consequently the higher-homotopy variable remains additive, while the fundamental-group variable is generally not. If the action is trivial the mixed product is zero. This explains exactly where the usual bilinearity statement fails.

Put $m=2n$ and let $\alpha$ be the identity class of $S^m$. With the product-cell orientation used in part (i), form the folding map $\nabla :S^m\vee S^m\to S^m$, which is identity on each summand. The top cell of $S^m\times S^m$ is attached by the product of the two inclusions, so folding gives the attaching map $[\alpha ,\alpha {]}_W$. Thus there is a cellular mapwhich is the fold on the lower skeleton and degree one on the top relative cell.

Let $a,b\in H^m(S^m\times S^m;\mathbb{Z})$ be the classes from the two factors, and let $x\in H^m(C;\mathbb{Z})$ restrict to the generator of the bottom sphere. Orient the top class $z\in H^{2m}(C;\mathbb{Z})$ so that $F^{\ast }z=ab$. Then $F^{\ast }x=a+b$. Since $a^2=b^2=0$ and $m$ is even, graded commutative algebra givesThe map $F^{\ast }$ is an isomorphism in degree $2m$, because of its degree-one top-cell map. Therefore $x^2=2z$. By the definition of the Hopf invariant this provesThe shifted Lie convention in part (i) multiplies the geometric bracket by $(-1{)}^m=1$, so it gives the same square and the same invariant. Reversing the orientation chosen for the attached top cell would reverse both its generator and the numerical sign convention; with the standard orientation the value is the stated positive two.

Use shifted degreesIts graded Lie bracket is the sign-adjusted Whitehead bracket from Question 2, equivalently the Samelson product on loops. This regrading is essential: the geometric bracket has degree minus one in the unshifted sphere grading. A free graded Lie algebra on a graded vector space $V$ means $\mathbb{L}(V)$ with the universal extension property for linear maps from $V$ to any graded Lie algebra.

Here are the general results used. The Bott–Samelson theorem says that, for a connected based CW complex $X$, the inclusion $X\to \Omega \Sigma X$ induces an isomorphism of graded algebraswith the Pontryagin product. The Milnor–Moore theorem says that for simply connected $Y$, rational loop Hurewicz homomorphism identifies $L_Y$ with the primitives of that loop-homology Hopf algebra and induces $U(L_Y)\cong H_{\ast }(\Omega Y;\mathbb{Q})$. The graded Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra of $U(L)$, for its word-length filtration, with the graded symmetric algebra on $L$; hence its primitive elements of a Hopf algebra are exactly $L$ over $\mathbb{Q}$. Finally the rational Whitehead theorem identifies rational homology equivalences of simply connected CW complexes with rational homotopy equivalences. Rationalization of a topological space induces the corresponding isomorphisms of loop-space homotopy and rational homology.

Suppose first $Y$ has the rational homotopy type of $\Sigma X$. A reduced suspension is simply connected when $X$ is connected; a disconnected $X$ has $H_1(\Sigma X;\mathbb{Q})={\tilde{H}}_0(X;\mathbb{Q})\ne 0$, so it cannot have the rational homology of the simply connected $Y$. Put $V={\tilde{H}}_{\ast }(X;\mathbb{Q})$, in strictly positive degrees. Bott–Samelson theorem gives $H_{\ast }(\Omega Y;\mathbb{Q})\cong T(V)$ as an algebra. To deduce freeness of the primitive Lie algebra, its coproduct must also be considered: the initial Bott–Samelson theorem generators can have reduced-diagonal terms when $X$ has nontrivial cup products.

By Milnor–Moore theorem write this Hopf algebra as $U(L_Y)$. Its algebra indecomposables satisfyThe first identity follows from the enveloping relation: after all positive products are killed, every bracket is killed and no additional linear relation is introduced. Hence each homogeneous basis vector of $V$ has a primitive representative in $H_{\ast }(\Omega Y;\mathbb{Q})$. Choose such representatives $p_v$. Sending $v$ to $p_v$ defines an algebra map from $T(V)$ to itself whose effect on indecomposables is the identity. It is an isomorphism: $p_v=v+$ decomposable terms, and every factor in those terms has smaller positive degree than $v$. Degree induction solves recursively for each original generator and constructs the inverse. This works even with arbitrarily many generators in a degree.

Equip the source tensor algebra with primitive generators. The map is now a Hopf-algebra isomorphism, because the chosen images are primitive. The primitives in this tensor Hopf algebra are exactly $\mathbb{L}(V)$: $U(\mathbb{L}(V))=T(V)$ by the two universal properties, and the graded Graded Poincaré–Birkhoff–Witt theorem gives its primitive subspace. Consequently $L_Y\cong \mathbb{L}(V)$, proving the forward direction without assuming a primitive coproduct for the original generators.

Conversely suppose $L_Y\cong \mathbb{L}(W)$ for a positively graded $W$. Choose a homogeneous basis $w_{\lambda }$ of $W$. Represent its corresponding homotopy class by a based sphere map $S^{|w_{\lambda }|+1}\to Y_{\mathbb{Q}}$, and combine these maps intoThe wedge sum is a reduced suspension, namely $Z=\Sigma ({\bigvee }_{\lambda }{S}^{|w_{\lambda }|})$. The preceding direction, applied to this wedge sum, gives its free rational homotopy Lie algebra on the sphere inclusions. Indeed these particular tensor generators are primitive because the reduced diagonal of each sphere is zero, and loop Hurewicz homomorphism sends the adjoint sphere inclusion to that generator. Naturality of the bracket shows that $F_{\ast }$ sends the free generating basis to the prescribed free generating basis of $L_Y$. It is therefore an isomorphism of free graded Lie algebras, hence an isomorphism in every rational homotopy degree. The rational Whitehead theorem makes $F$ a rational homotopy equivalence. This proves the reverse implication and also shows that a wedge sum of spheres can be chosen as the reduced suspension.

For every chosen connected $X$ with $Y{\simeq }_{\mathbb{Q}}\Sigma X$, the required algebra statement is precisely the Bott–Samelson theorem isomorphism, transported along rational homotopy equivalence:Multiplication is concatenation of tensors and the degree-zero tensor is the unit. All freeness assertions here are graded, with the Koszul signs described above.

Let $Z={\mathbb{CP}}^5/{\mathbb{CP}}^2$. Its quotient cell structure has one cell in each of dimensions $0,6,8,10$. In particular $Z$ is five-connected. The relative cohomology sequence, or its cellular cochain complex, givesThe subscripts denote degrees. Under the quotient map back to ${\mathbb{CP}}^5$, the three positive classes pull back to $x^3,x^4,x^5$. All positive products vanish, since their degree is at least twelve. Thus the cohomology ring of a collapsed projective subspace in this case is the square-zero algebra $R$ on these three classes.

Now compute its Sullivan minimal model $(\Lambda V,d)$, using cohomological grading and the convention that even generators generate a polynomial ring and odd generators an exterior algebra. Five-connectivity gives $V^j=0$ for $j\le 5$. Since a minimal differential is decomposable, it cannot have a nonzero value below degree twelve. Matching cohomology through degree ten therefore forceswith no other generators in degrees at most ten. The first products must then be killed by generatorsThese follow by killing respectively the unwanted degree-twelve, degree-fourteen, and two degree-sixteen classes. Subsequent generators kill $bc,c^2$ and the resulting syzygies. For example $av-bu$ is closed of degree nineteen, so it requires a further generator of degree eighteen. Thus the full minimal model is not just the algebra on the first seven generators.

There is nevertheless a simple map from the full model to $R$: send $a,b,c$ to the three displayed classes and every other generator to zero. It is a chain map, since every generator differential is a sum of products of positive-degree elements, and all positive products in $R$ vanish. It induces an isomorphism in cohomology: the entire cohomology of $\Lambda V$ is concentrated in degrees $0,6,8,10$, and its indicated basis maps to the indicated basis of $R$. This proves formality of a topological space and identifies the full minimal model as a minimal resolution of $R$, rather than only matching the initial part of its cohomology.

An explicit description of that full resolution is useful. Take the free graded Lie algebraIts Chevalley–Eilenberg cochain algebra $C^{\ast }(L)$ has one generator of degree $|\ell |+1$ for every homogeneous Lie basis word $\ell$, and its quadratic differential is dual to the bracket, with the usual graded signs. Rescaling the bracket-word duals gives exactly the generators and differentials listed above. Positive Lie degrees make this algebra Sullivan minimal. Its cohomology is $R$: the universal enveloping algebra is $T(p,q,r)$, and the augmentation module has the exact free resolutionExactness is immediate because every nonempty tensor word has a unique last letter. Applying ${\mathrm{Hom}}_{T(p,q,r)}(-,\mathbb{Q})$ gives just the unit and the three dual classes in total degrees $6,8,10$; products of positive classes land in an absent Ext degree two and are zero. Chevalley–Eilenberg cochains of a graded Lie algebra compute this Ext, so this also verifies the full model's cohomology.

The wedge sum $W=S^6\vee S^8\vee S^{10}$ has the same connectivity, cohomology ring, and initial closed minimal generators. The same full-model-to-$R$ argument applies to $W$, so its minimal model is also $C^{\ast }(L)$, by uniqueness of Sullivan minimal models of quasi-isomorphic simply connected rational cochain algebras. The Sullivan minimal-model classification identifies isomorphic minimal models with the same rational homotopy type. ConsequentlyThe explicit infinite minimal resolution above is shared by both spaces.

It is enough to assume that $h_k:{\pi }_k(X)\otimes \mathbb{Q}\to H_k(X;\mathbb{Q})$ is surjective for every $k\ge 2$; an integrally surjective Hurewicz homomorphism certainly has this property. First give the minimal-model argument suggested by the hint. Let $A=(\Lambda V,d)$ be a simply connected Sullivan minimal model, with $V$ in degrees at least two. Put $A^{+}={⨁}_{k>0}{A}^k$. Its indecomposable quotient of an augmented algebra isMinimality makes the induced differential on $V$ zero, so projection is a chain map $q:A^{+}\to (V,0)$. The Sullivan model identification says that the induced mapis dual to rational Hurewicz homomorphism. Under the usual finite-type duality convention, the assumed surjectivity makes $q_{\ast }$ injective.

A product of two positive cohomology classes has a decomposable cocycle representative, so its image under $q_{\ast }$ is zero. Injectivity therefore givesThis deduction uses the hypothesis; zero cup products by themselves would not yet prove formality of a topological space.

Choose a homogeneous cohomology basis $h_{\lambda }$ and cocycle representatives $z_{\lambda }\in A$. Their indecomposable parts $q(z_{\lambda })$ are linearly independent. Extend them to a graded basis of $V$ and replace the corresponding generators by the actual cocycles $z_{\lambda }$. This is a valid change of free algebra generators: each $z_{\lambda }$ differs from its indecomposable part by products of generators of strictly smaller degree, so degree induction constructs the inverse substitution. Call these closed generators $C$, and the remaining generators $N$.

Define a map of graded algebrasIts target has zero differential. Every differential in $A$ is decomposable, and all products of positive target classes vanish, so $\varphi$ is a chain map. If $z$ is any positive cocycle, its indecomposable part is in the span of the $q(z_{\lambda })$, because these form the image of $q_{\ast }$. Write $q(z)={\sum }_{\lambda }{c}_{\lambda }q(z_{\lambda })$. Then $\varphi (z)={\sum }_{\lambda }{c}_{\lambda }{h}_{\lambda }$, while injectivity of $q_{\ast }$ gives $[z]={\sum }_{\lambda }{c}_{\lambda }{h}_{\lambda }$. Thus $\varphi$ induces the identity on cohomology and is a quasi-isomorphism. This proves that $X$ is a formal space.

The rational wedge sum assertion has a direct topological proof, which also avoids finite-dimensional duality restrictions in the preceding model argument. Choose a homogeneous basis $b_{\lambda }$ of ${\tilde{H}}_{\ast }(X;\mathbb{Q})$. By surjectivity, each basis class is the image under the Hurewicz homomorphism of an element of ${\pi }_{n_{\lambda }}(X)\otimes \mathbb{Q}$. In $X_{\mathbb{Q}}$ this element is represented by an actual sphere map $S^{n_{\lambda }}\to X_{\mathbb{Q}}$. Simple connectivity and $H_1=0$ ensure $n_{\lambda }\ge 2$. The wedge sum of these maps givesand induces an isomorphism of rational homology by the choice of basis. The rational Whitehead theorem now makes it a rational homotopy equivalence, givingA wedge sum of spheres is a formal space. For each individual sphere, its augmented minimal model maps by a quasi-isomorphism to its sphere cochain algebra, and also to its cohomology algebra, by killing the extra odd generator in the even-sphere case. Take based simplicial models of the spheres. Their wedge sum is a simplicial pushout over the common basepoint, so the algebra of rational polynomial differential forms is the corresponding fibre product of augmented sphere algebras. Taking fibre products of these model maps gives a quasi-isomorphism zigzag to the square-zero cohomology algebra. This also works for infinitely many sphere summands: the positive parts are products of the individual complexes, and products of vector-space exact sequences over a field are exact. In the finite case its minimal resolution is the minimal model of a wedge of simply connected spheres. Thus the topological proof establishes both conclusions without imposing finite rank on the sphere basis. For the zero reduced-homology case the wedge sum is a point. Both the minimal-model construction and the explicit sphere map explain why the surjectivity hypothesis is decisive.

Use cohomological degree, and write the Sullivan minimal models asandThe subscripts indicate degrees. The even generators generate a polynomial ring and the odd generators an exterior algebra. Their cohomology rings are respectively $\mathbb{Q}[x]/(x^{n+1})$ and $\mathbb{Q}[u]/(u^2)$, since each odd differential kills exactly the indicated non-zero-divisor polynomial relation. These are the usual projective-space and even-sphere minimal models.

Orient the top classes so that the degree-one condition is $f^{\ast }[u]=[x^n]$. A representative model map must send $u$ to $x^n$. It must send $v$ to an element of degree $4n-1$ with differential $x^{2n}$. The element $x^{n-1}y$ has exactly these properties. ThusThe differential check is $d(x^{n-1}y)=x^{n-1}{x}^{n+1}=x^{2n}=(f^{\ast }u{)}^2$. In these degrees there is no other available monomial: even powers of $x$ have even degree, and an odd monomial is a power of $x$ times $y$. Hence the coefficient is forced by the differential. For $n=1$ the two models coincide and the map is the identity model map.

The homotopy fibre is the homotopy pullback of the path fibration over the target sphere along $f$. The Sullivan fibre-model theorem computes it by taking a relative acyclic model of that path fibration and tensoring it over the target model with the source model. Here the construction can be written out and checked directly.

Adjoin $a$ of degree $2n-1$ and $b$ of degree $4n-2$ to the sphere model, withThis is a differential, because $D(v-au)=u^2-u^2=0$. It is an acyclic relative Sullivan algebra: putting $v^{\prime }=v-au$ makes its two pairs $Da=u$ and $Db=v^{\prime }$. The first is a polynomial generator killed by an odd generator; the second is an odd generator killed by an even polynomial generator. Each pair has cohomology only $\mathbb{Q}$, the second by the ordinary polynomial antiderivative calculation in characteristic zero. Hence this models the contractible path space, and its augmentation models the chosen basepoint.

Tensoring with the projective-space model via the map in part (i) givesThe last differential is necessary: using only $Db=f^{\ast }v$ would give $D^{2}b=x^{2n}\ne 0$. Make the invertible generator change $z=y-xa$, of degree $2n+1$. Since $x$ is even,For $n\ge 2$, all nonzero differentials are decomposable and the generator ordering $x,a,z,b$ is Sullivan. Thus no further minimization is needed:For example at $n=2$ this is $(\Lambda (x_2,a_3,z_5,b_6),Da=x^2,Db=xz)$.

The indecomposable generators of a simply connected finite-type minimal model are dual to its rational homotopy groups. Therefore, for $n\ge 2$,These four degrees are distinct. The exact homotopy sequence provides an independent check: ${\mathbb{CP}}^n$ has rational homotopy only in degrees $2,2n+1$, while $S^{2n}$ has it in degrees $2n,4n-1$. For $n\ge 2$ these source and target degrees do not overlap, so the source classes survive in the fibre and the target classes contribute their one-degree-lower connecting classes, giving the same four groups.

At $n=1$, the displayed relative fibre algebra is not minimal: $Da=x$ and $Db=z$ are linear pairs. Cancelling both acyclic pairs leaves $\mathbb{Q}$. AccordinglyThis agrees with ${\mathbb{CP}}^1\cong S^2$ and the degree-one map being a homotopy equivalence. The case distinction is required before reading homotopy groups from the generators.