Rational Cohomology Algebra Research Articles

Equivariant formality of the isotropy action on (ℤ2 ⊕ ℤ2)-symmetric spaces

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalization, it is even known that their isotropy action is equivariantly formal. In this paper, we show that [Formula: see text]-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

Cohomology algebra of mapping spaces between quaternion Grassmannians

Let Gk,n(ℍ) for 2≤k<n denote the quaternion Grassmann manifold of k-dimensional vector subspaces of ℍn. In this paper we compute, in terms of the Sullivan models, the rational cohomology algebra of the component of the inclusion i: Gk,n(ℍ) → Gk,n+r(ℍ) in the space of mappings from Gk,n(ℍ) to Gk,n+r(ℍ) for r≥1 and, more generally, we show that the cohomology of Map(Gk,n(ℍ),Gk,n+r(ℍ);i) contains a truncated algebra ℚ[x]x4r+n+k^{2}-nk for n≥4.

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

We consider the complex Grassmannian Grk,n of k-dimensional subspaces of ℂn. There is a natural inclusion in,r:Grk,n↪Grk,n+r. Here, we use Sullivan models to compute the rational cohomology algebra of the component of the inclusion in,r in the space of mappings from Grk,n to Grk,n+r for r≥1 and in particular to show that the cohomology of mapGrn,k,Grn,k+r;in,r contains a truncated algebra ℚx/xr+n+k2−nk, where x=2, for k≥2 and n≥4.

Two examples of toric arrangements

We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determine by the arithmetic matroid (however it is determine by the poset of layers).

The spectrum for commutative complex K–theory

We study commutative complex $K$-theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex $K$-theory is stably equivalent to the $ku$-group ring of $BU(1)$ and thus obtain a splitting of its representing space $B_{com}U$ as a product of all the terms in the Whitehead tower for $BU$, $B_{com}U\simeq BU\times BU\langle 4\rangle \times BU\langle 6\rangle \times \dots .$ As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for $B_{com}U$ we describe the relationship of our results with a previous computation of the rational cohomology algebra of $B_{com}U$. This gives an essentially complete description of the space $B_{com}U$ introduced by A. Adem and J. G\'omez.

The Borel cohomology of the loop space of a homogeneous space

Let B′→fB←pE be a diagram in which p is a fibration and the pair (f,p) of the maps is relatively formalizable. Then, we show that the rational cohomology algebra of the pullback of the diagram is isomorphic to the torsion product of algebras H⁎(B′) and H⁎(E) over H⁎(B). Let M be a space which admits an action of a Lie group G. The isomorphism of algebras enables us to represent the cohomology of the Borel construction of the space of free (resp. based) loops on M in terms of the torsion product if M is equivariantly formal (resp. G-formal). Moreover, we compute explicitly the S1-equivariant cohomology of the space of the based loops on the complex projective space CPm, where the S1-action is induced by a linear action of S1 on CPm.

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.

The center of a rational cohomology algebra for some loop spaces with respect to the Pontryagin product

We solve the problem of calculation of the center of the rational cohomology algebra H*(ΩX;ℚ) of the loop space of simply connected four-dimensional manifolds with the Pontryagin product. The well-known results of Milnor and Moore represent H*(ΩX) as a universal enveloping Lie algebra π*(ΩX) endowed with the Whitehead-Samelson bracket. Neisendorfer obtained a representation of the algebra Π*(ΩX) ⊗ ℚ in terms of generators and relations for the case when X is a simply connected four-dimensional manifold. In this paper, the center Z(H*(ΩX;ℚ)) is calculated using this representation.

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.

Iterated cyclic homology

From the viewpoint of rational homotopy theory, we introduce an iterated cyclic homology of connected commutative differential graded algebras over the rational number field, which is regarded as a generalization of the ordinary cyclic homology. Let T be the circle group and $\mathscr F$ (Tl, X) denote the function space of continuous maps from the l-dimensional torus Tl to an l-connected space X. It is also shown that the iterated cyclic homology of the differential graded algebra of polynomial forms on X is isomorphic to the rational cohomology algebra of the Borel space ET × T $\mathscr F$ (Tl, X), where the T-action on $\mathscr F$ (Tl, X) is induced by the diagonal action of T on the source space Tl.

The rational topology of gauge groups and of spaces of connections

Let P be a principal bundle with semisimple compact simply connected structure group G over a compact simply connected four-manifold M. In this paper we give explicit formulas for the rational homotopy groups and cohomology algebra of the gauge group and of the space of (irreducible) connections modulo gauge transformations for any such bundle.

Evaluation fibrations and topology of symplectomorphisms

There are two main results. The first states that isotropy subgroups of groups acting transitively on rationally hyperbolic spaces have infinitely generated rational cohomology algebra. Using this fact, we prove that the analogous statement holds for groups of symplectomorphisms of certain blowups.

Note on the rational cohomology of the function space of based maps

In this paper, for a formal, path connected, finite-dimensional CW-complex X of finite type and a q-connected space Y of finite type with q > dimX, we determine the necessary and sucient condition for the rational cohomology algebra H ⁄ (F⁄(X;Y );Q) of the function space F⁄(X;Y ) of based maps to be free.

Vassiliev invariants and the cubical knot complex

We construct a cubical CW-complex CK( M) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M. We construct CK( R 3) by attaching cells to CK( R 3) for every degenerate 1-singular and 2-singular knot, and we show that π 1( CK( R 3))=1 and π 2( CK( R 3))= Z . We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R 3 we show that v 2 uniquely generates H 2( CK, D), where D is the subcomplex of degenerate singular knots. More generally, we show that any Vassiliev invariant coming from the Conway polynomial is nontrivial in cohomology. The cup product in H ∗(CK) provides a new graded commutative algebra of Vassiliev invariants evaluated on ordered singular knots. We show how the cup product arises naturally from a cocommutative differential graded Hopf algebra of ordered chord diagrams.

Rational Betti numbers of configuration spaces

We compute the rational Betti numbers of the configuration space C k(M) of k points in an even-dimensional orientable closed manifold M and prove that these numbers depend only on the rational cohomology algebra of the manifold. We give also a formula for the Euler–Poincaré characteristic of C k(M) in terms only of k, and of the Euler–Poincaré characteristic of M.

Postnikov sections of formal and hyperformal spaces

Let X be a simply connected CW complex and X n {X_n} its nth Postnikov section. We prove that X is formal provided H q ( X n , Q ) {H^q}({X_n},\mathbb {Q}) is additively generated by decomposables for all q and n with q > n q > n . Recall from [4] that a space X is said to be hyperformal if its rational cohomology algebra is the quotient of a free graded algebra by an ideal generated by a regular sequence. Using the main result of Felix and Halperin’s paper (Trans. Amer. Math. Soc. 270 (1982), 575-588) we show our sufficient condition for formality is actually equivalent to hyperformality.

On a generic property of geodesic flows

In this paper we are concerned with a sufficient condition for a Riemannian metric on a compact simply–connected manifold to have infinitely many geometrically distinct closed geodesics. 1969 Gromoll–Meyer proved in [GM] that for every Riemannian metric on a compact simply–connected manifold M there are infinitely many geometrically distinct closed geodesics if the sequence (bi(ΛM ;F ))i of Betti numbers of the free loop space ΛM is unbounded for some field F . Using the theory of minimal models Vigue-Poirrier/Sullivan proved that the rational cohomology algebra H∗(M ; I Q) of M is generated by a single element if and only if the sequence (bi(ΛM ; I Q))i of Betti numbers of the free loop space ΛM of M is bounded. It is a conjecture that the same statement holds for all fields of prime characteristic, partial results are due to McCleary–Ziller [MZ] and Halperin/Vigue-Poirrier [HV]. Now we turn to the manifolds for which the hypothesis of Gromoll– Meyer’s theorem does not hold, for example spheres and projective spaces. Then stability properties of the closed geodesics become important. A closed geodesic is hyperbolic if the linearized Poincare map has no eigenvalue of norm 1. From the bumpy metrics theorem due to Abraham [Ab] and Anosov [An2] and from a pertubation result due to Klingenberg–Takens [KT] one can conclude: A C4–generic metric on a compact manifold has either a non–hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic. In the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by Moser [Mo]. In the second case there are infinitely many closed geodesics if M is simply–connected due to results by Hingston [Hi] and the author [Ra1]. Hence it follows from these results that a C4–generic metric on a compact simply–connected manifold has infinitely many geometrically distinct closed geodesics. We remark that it is an open question whether there is a Riemannian metric on a simply–connected compact manifold all of whose closed geodesics are hyperbolic. In [Ra2] we show that the examples of metrics on the 2–

Rational Homotopy Types with the Rational Cohomology Algebra of Stunted Complex Projective Space

AbstractWe consider the number of spaces, up to rational homotopy equivalence, which have rational cohomology algebra isomorphic to that of stunted complex projective space . Using a classification theory due to Schlessinger and Stasheff, we determine the number of rational homotopy types with rational comology algebra isomorphic to , for any given n and k. The necessary computations make use of a spectral sequence introduced by the second named author.

The set of multiplicative predifferentials and the rational cohomology algebra of fibre spaces

For a given commutative graded algebra H over a field K of characteristic zero the contravariant functor D n (∵ H) from the category of topological spaces to the category of pointed sets is defined. To a Serre fibration F → E → X the element d(ξ) ∈ D n (X:H). H = H(F: K) , is functorially assigned which determines a new model for ξ. Moreover, the description of the set [ X, Y] 0 is given.