Transversality for the moduli space of Spin(7)-instantons

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.

On the Cohomology of the Classifying Spaces of Projective Unitary Groups

Let [Formula: see text] be the classifying space of [Formula: see text], the projective unitary group of order [Formula: see text], for [Formula: see text]. We use a Serre spectral sequence to determine the ring structure of [Formula: see text] up to degree [Formula: see text], as well as a family of distinguished elements of [Formula: see text], for each prime divisor [Formula: see text] of [Formula: see text]. We also study the primitive elements of [Formula: see text] as a comodule over [Formula: see text], where the comodule structure is given by an action of [Formula: see text] on [Formula: see text] corresponding to the action of taking the tensor product of a complex line bundle and an [Formula: see text]-dimensional complex vector bundle.

A REMARK ON THE STABLE REAL FORMS OF COMPLEX VECTOR BUNDLES OVER MANIFOLDS

Let$M$be an$n$-dimensional closed oriented smooth manifold with$n\equiv 4\;\text{mod}\;8$, and$\unicode[STIX]{x1D702}$be a complex vector bundle over$M$. We determine the final obstruction for$\unicode[STIX]{x1D702}$to admit a stable real form in terms of the characteristic classes of$M$and$\unicode[STIX]{x1D702}$. As an application, we obtain the criteria to determine which complex vector bundles over a simply connected four-dimensional manifold admit a stable real form.

Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles

Using a modification of Webster's proof of the Newlander–Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

Splitting ℂP∞ and Bℤ/pn into Thom spectra

In recent years, much work in algebraic topology has been devoted to stable splitting phenomena. Often the existence of these splittings has first been detected at the cohomological level in terms of modules over the Steenrod algebra.For example, W. Richter has exhibited a decomposition of ΩSU(n) of the form(see [7]). Not only were cohomology calculations the initial evidence for this situation, but they further suggested that each summand Gk might be the Thom complex of a suitable k-plane complex vector bundle. This possibility was also verified by Mitchell.

On Chern classes of stably fibre homotopic trivial bundles

Let ξ be a stably fibre homotopic trivial vector bundle. A classical result of Thorn states that the Stiefel-Whitney classes of ξ vanish, and one way to prove this is as follows. Letube the Thorn class of ξ in mod 2 cohomology. Thenuis stably spherical by [2] and therefore all stable cohomology operations vanish onu, showing thatwi(ξ)u= Sqiu= 0. In this note we shall apply this same method using complex cobordism and Landweber-Novikov operations to study relations among Chern classes of a stably fibre homotopic trivial complex vector bundle. We will thus obtain in a unified way certain strong modpconditions for every primep.

On complex vector bundles on rational threefolds

It is known [14] that every topological complex vector bundle on a smooth rational surface admits an algebraic structure. In [10] one constructs algebraic vector bundles of rank 2 on with arbitrary Chern classes c1, c2 subject to the necessary topological condition c1 c2 = 0 (mod 2). However, in dimensions greater than 2 the Chern classes don't determine the topological type of a vector bundle. In [2] one classifies the topological complex vector bundles of rank 2 on and one proves that any such bundle admits an algebraic structure.

First Chern class and holomorphic tensor fields

Let M be an n-dimensional compact Kaehler manifold, TM its (holomorphic) tangent bundle and T*M its cotangent bundle. Given a complex vector bundle E over M, we denote its m-th symmetric tensor power by SmE and the space of holomorphic sections of E by Γ(E).

On the Kunneth formula spectral sequence in equivariant K- theory

Let G be a compact, connected Lie group such that π2(G) is torsion free. Throughout this paper a vector bundle (representation) will mean a complex vector bundle (representation) and KG will denote the equivariant K-theory functor associabed with the group, G.