PL Characteristic Classes and Cobordism (original) (raw)

Explicit PL Self-Knottings and the Structure of PL Homotopy Complex Projective Spaces

2015

We show that certain piecewise-linear homotopy complex projective spaces may be described as a union of smooth manifolds glued along their common boundaries. These boundaries are sphere bundles and the glueing homeomorphisms are piecewise-linear self-knottings on these bundles. Furthermore, we describe these self-knottings very explicitly and obtain information on the groups of concordance classes of such maps. A piecewise linear homotopy complex projective space CPn is a compact PL manifold M2n homotopy equivalent to CPn. In [22] Sullivan gave a complete enumeration of the set of PL isomorphism classes of these manifolds as a consequence of his Characteristic Variety theorem and his analysis of the homotopy type of G/PL. In [15] Madsen and Milgram have shown that these manifolds, the index 8 Milnor manifolds, and the differentiable generators of the oriented smooth bordism ring provide a complete generating set for the torsion-free part of the oriented PL bordism ring. Hence a stud...

A Note on the Algebra of Operations for Hopf Cohomology at Odd Primes

2017

Let p be any prime, and let B(p) be the algebra of operations on the cohomology ring of any cocommutative F_p-Hopf algebra. In this paper we show that when p is odd (and unlike the p=2 case), B(p) cannot become an object in the Singer category of F_p-algebras with coproducts, if we require that coproducts act on the generators of B(p) coherently with their nature of cohomology operations

On Pointed Ribbon Hopf Algebras

Journal of Algebra, 1996

In his paper, ''On Kauffman's knot Invariants Arising from Finite w x Dimensional Hopf Algebras'' R1 , Radford constructed two extensive families of pointed Hopf algebras. The first one, denoted by H , n, q, N, generalizes Sweedler's well known 4-dimensional noncommutative and noncocommutative Hopf algebra. The second one, denoted by U , is Ž N, ,. a family of finite dimensional pointed unimodular ribbon Hopf algebras constructed for the purpose of computing knots and 3-manifold invariants. Ž. X This family generalizes the well known quantum group U sl where q is q 2 * This work was partially supported by the Basic Research Foundation administrated by the Israel Academy of Sciences and Humanities. This work was partially done while the author was at U.I.C. and he thanks them for their warm hospitality. This is part of the author's Ph.D. thesis. 760

Hopf quasigroups and the algebraic 7-sphere

Journal of Algebra, 2010

We introduce the notions of Hopf quasigroup and Hopf coquasigroup H generalising the classical notion of an inverse property quasigroup G expressed respectively as a quasigroup algebra kG and an algebraic quasigroup k[G]. We prove basic results as for Hopf algebras, such as anti(co)multiplicativity of the antipode S : H → H, that S 2 = id if H is commutative or cocommutative, and a theory of crossed (co)products. We also introduce the notion of a Moufang Hopf (co)quasigroup and show that the coordinate algebras k[S 2 n −1 ] of the parallelizable spheres are algebraic quasigroups (commutative Hopf coquasigroups in our formulation) and Moufang. We make use of the description of composition algebras such as the octonions via a cochain F introduced in [2]. We construct an example k[S 7 ] ⋊ Z 3 2 of a Hopf coquasigroup which is noncommutative and non-trivially Moufang. We use Hopf coquasigroup methods to study differential geometry on k[S 7 ] including a short algebraic proof that S 7 is parallelizable. Looking at combinations of left and right invariant vector fields on k[S 7 ] we provide a new description of the structure constants of the Lie algebra g 2 in terms of the structure constants F of the octonions. In the concluding section we give a new description of the q-deformation quantum group Cq[S 3 ] regarded trivially as a Moufang Hopf coquasigroup (trivially since it is in fact a Hopf algebra) but now in terms of F built up via the Cayley-Dickson process.

On the cohomology of some Hopf algebroids and Hattori-Stong theorems

Homology, Homotopy and Applications, 2000

We apply group cohomological methods to calculate the cohomology of K(n) * BP as a K(n) * K(n)-comodule, recovering recent results of Hovey and Sadofsky. As applications we determine the Chromatic Spectral Sequence for BP based on Johnson and Wilson's E(n), showing the relationship to some generalizations of the classical Hattori-Stong Theorem and determine the change of Hopf algebroid spectral sequence associated with the natural map BP −→ E(n), extending calculations of Clarke for the Todd orientation M U −→ KU .

Homotopy and homology of simplicial abelian Hopf algebras

Mathematische Zeitschrift, 1999

Let A be a simplicial bicommutative Hopf algebra over the eld F 2 with the property that 0 A = F 2 . We show that A is a functor of the Andr e-Quillen homology of A, where A is regarded as an F 2 algebra. Then we give a method for calculating that Andr e-Quillen homology independent of knowledge of A. Let G be an abelian group. Since the work of Serre 19] and Cartan 6], we have known that the mod p homology of an Eilenberg-MacLane space K(G; n), n 1, depends only on Tor s (Z=p; G), s = 0; 1. More is true: the structure of H K(G; n) = H (K(G; n); F p ) as an unstable coalgebra over the Steenrod algebra depends only on the Tor groups and the Bockstein : Tor 1 (Z=p; G) ! Tor 0 (Z=p; G) = Z=p G which is the connecting homomorphism of the six term exact sequence obtained by tensoring G with the short exact sequence 0 ! Z=p ! Z=p 2 ! Z=p ! 0: The purpose of this paper to expand on this observation; indeed, our principal result will be that this is an algebraic fact, not a topological one, and an instance of a phenomenon that arises naturally in the study of simplicial bicommutative Hopf algebras. With an appropriate model for an Eilenberg-MacLane space { for example, the simplicial abelian group model { the mod p homology groups of the Eilenberg-MacLane space K(G; n) are the homotopy groups of the simplicial bicommutative Hopf algebra F p K(G; n); that is, H K(G; n) = H (K(G; n); F p ) = F p K(G; n): Since K(G; n) is connected (as n > 0), we have that 0 F p K(G; n) = F p ; we will say that the simplicial Hopf algebra F p K(G; n) is homotopy connected. We will prove, at least when 1 The rst author was partially supported by the NSF.