Rational Cohomology Groups Research Articles

Hodge-theoretic splitting mechanisms for projective maps

According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three dierent splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we use the relative hard Lefschetz theorem and elementary linear algebra to construct ve distinct splittings, two of which seem to be new, and to prove that they are splittings of mixed Hodge structures.

Representation stability for the cohomology of the pure string motion groups

The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups given by Jensen, McCammond, and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group vanish in positive degree. We also prove that the subgroup of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine \({H^k\left({\bar S}_{g}, {\mathbb Q}\right)}\) for g ≥ 2 and k ≤ 3, where \({{\bar S}_{g}}\) denotes the moduli space of spin curves of genus g.

Motivic Landweber exactness

We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landweber-type formula involving the \mathsf{MGL} -homology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal coefficient spectral sequence we deduce formulas for operations of certain motivic Landweber exact spectra including homotopy algebraic K -theory. Finally we employ a Chern character between motivic spectra in order to compute rational algebraic cobordism groups over fields in terms of rational motivic cohomology groups and the Lazard ring.

Rational homotopy of gauge groups

In this brief paper, we observe that basic results from rational homotopy theory provide formulas for the rational homotopy groups of gauge groups of principal bundlesK→P→BK \to P \to Bin terms of the rational homotopy groups ofKKand cohomology groups ofBBalone.

The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group

In this paper, we determine the cokernel of the k-th Johnson homomorphisms of the automorphism group of a free metabelian group for k > 2 and n > 4. As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is isomorphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus representations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map.

Banach algebras and rational homotopy theory

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GLn(A), the group of invertible n × n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lcn(A) denote the space of “last columns” of GLn(A). We construct a natural isomorphism Ȟ(Max(A);Q) ∼= π2n−1−s(Lcn(A))⊗ Q for n > 1 2 s+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F (X,G) for G a Lie group or, more generally, a rational H-space.

The Cohomology of Real De Concini–Procesi Models of Coxeter Type

We study the rational cohomology groups of the real De Concini–Procesi model corresponding to a finite Coxeter group, generalizing the type-A case of the moduli space of stable genus 0 curves with marked points. We compute the Betti numbers in the exceptional types, and give formulae for them in types B and D. We give a generating-function formula for the characters of the representations of a Coxeter group of type B on the rational cohomology groups of the corresponding real De Concini–Procesi model, and deduce the multiplicities of one-dimensional characters in the representations, and a formula for the Euler character. We also give a moduli space interpretation of this type-B variety, and hence show that the action of the Coxeter group extends to a slightly larger group.

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 11.

The Johnson homomorphism and the third rational cohomology group of the Torelli group

Using the Johnson homomorphism and the representation theory of the symplectic group, we study the third rational cohomology group of the Torelli group, which gives characteristic classes of surface bundles whose holonomy groups act trivially on the homology groups of their fibers.

A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres

We give an elementary proof of the rational Hurewicz theorem and compute the rational cohomology groups of Eilenberg–MacLane spaces and the rational homotopy groups of spheres. Instead of using the Serre spectral sequence, we only assume the classical Hurewicz theorem, and give a short proof of the rational Gysin and Wang long exact sequences, which are applied inductively to the path fibration of Eilenberg–MacLane spaces.

Rational isomorphisms between K-theories and cohomology theories

The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the “Segre map”) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the “topological filtration” on singular cohomology groups in K-theoretic terms.

Some calculations of cohomology groups of finite Alexander quandles

We calculate some quandle cohomology groups; the rational cohomology groups of any finite Alexander quandles, the second cohomology groups with a finite field coefficient of any finite Alexander quandles over a finite fields, and the third cohomology groups of the finite Alexander quandles of the form Z p[T,T −1]/(T−ω) .

Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

It is not known whether or not the stable rational cohomology groups $\tilde H^*(\operatorname{Aut}(F_\infty);\mathbb{Q})$ always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and $X_m$ of e;outer spacee: and e;auter spacee;, respectively, introduced in [3] by Culler and Vogtmann and [6] by Hatcher and Vogtmann.

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the “holomorphic K -theory” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in [11]. This theory is built out of studying algebraic bundles over a variety up to “algebraic equivalence”. In this paper we will give calculations of this theory for “flag like varieties” which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K theory and the appropriate “morphic cohomology” groups, defined in [7] in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the “symmetrized loop group” ΩU(n)/Σn where the symmetric group Σn acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K theory by inverting the Bott class, then rationally this is isomorphic to topological K theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K theory, and show that these obstructions vanish for generalized flag manifolds.

Finiteness of Minimal Modular Symbols for SLn

Let K/Q be a number field with ring of integers O, and let Γ⊂SLn(O) be a finite index subgroup. Using a classical construction from the geometry of numbers and the theory of modular symbols, we exhibit a finite spanning set of the highest nonvanishing rational cohomology group of Γ.

Calculating cohomology groups of moduli spaces of curves via algebraic geometry

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

On the cohomology of algebraic and related finite groups

In the above theorem, H*(GL,,M~ ~ denotes the rational cohomology of the algebraic group GL, with coefficients in M(, ~), the rational GL.-module obtained from the adjoint module M . = M ~ ~ through twisting by the r th power of the Frobenius endomorphism on GL,. The algebra structure on the cohomology is induced from the associative algebra structure on M,. On the other hand, we provide the following systematic determination of certain rational cohomology groups H*(G, V(r)), where V (r) is the rational Gmodule obtained from the rational G-module V by applying the r th Frobenius twist (see (1.2)). This is a somewhat sharpened version of Theorem 4 of [36].

Generic cohomology for twisted groups

Let G be a simple algebraic group defined and split over ko = Fp, and let a be a surjective endomorphism of G with finite fixed-point set Go. We give conditions under which cohomology groups of G are isomorphic to cohomology groups of Go. Let G be a simple algebraic group defined and split over k = Fp, and, for q = pm, let G(q) be the subgroup of Fq-rational points. For a finite dimensional rational G-module V and a nonnegative integer e, let V(e) be the G-module obtained by twisting the original G-action on V by the Frobenius endomorphism x > X1P of G. Cline, Parshall, Scott, and van der Kallen proved in [2] that, for sufficiently large q and e (depending on V and n), the restriction map induces an isomorphism from the rational cohomology group H'(G, V(e)) to H'(G(q), V(e)). This implies that, as q increases, the groups H'(G(q), V(e)) have a stable or generic value H en(G, V). In this paper, we prove an analogous theorem with G(q) replaced by Go for a surjective endomorphism a of G having finite fixed point set. The first section of the paper summarizes the basic results on endomorphisms of algebraic groups required for the proof. Some arithmetic facts are established in the second section, and the main theorem is proved in the third section. The author is grateful to Leonard L. Scott for several helpful conversations, and to the referee for comments on an earlier version of this paper. 1. Endomorphisms of algebraic groups. We briefly review here some results on endomorphisms of algebraic groups which will be needed later. We refer the reader to [4] and [5] for more details. Let k be the algebraic closure of the prime field ko = Fp and let G be a simple algebraic group defined and split over ko. If a is a surjective rational endomorphism of G having finite fixed-point set G0, a stabilizes a Borel subgroup B and a maximal torus T Ua is a T-equivariant isomorphism, then axa(u) = x (cauq(a)) for some Received by the editors May 5, 1980 and, in revised form, November 12, 1980. 1980 Mathematics Subject Classification Primary 20G10.

A theorem on the cohomology ring of nilpotent groups

Let G be a torsion free, finitely generated nilpotent group, and H be a subgroup of finite index of G. Mal'cev proved in [4] that G can be embedded in a connected, simply connected, nilpotent rational Lie group 65, such that G and H are discrete subgroups of 65 and the coset spaces 6/G and 6/H are compact. Nomizu showed in [5] that if L is the associated Lie algebra of 65, then Hn(L, Q+) -Hn(6/G, Q+) where Q+ is the additive group of rational numbers. Let Hn(G, Q+) denote the rational cohomology group of the abstract group G. It follows from Lie group theory that Hn(65/G, Q+)'t-Hn(G, Q+), hence Hn(L, Q+)--Hn(G, Q+), and for the same reason we have Hn(L, Q+) ,-Hn(H, Q+), whence Hn(G, Q+)>Hn(H, Q+). In this paper we shall, however, indicate a purely algebraic approach to this theorem in the following strengthened form: