Let F be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of F have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of F are a right module over a certain pro-operad. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor F can be reconstructed from this structure on the derivatives.

Stockholm University, Faculty of Science, Department of Mathematics.

Ching, Michael

MANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS2017In: Manifolds and K-Theory / [ed] Gregory Arone, Brenda Johnson, Pascal Lambrechts, Brian A. Munson, Ismar Volic, Providence: American Mathematical Society (AMS), 2017, p. 1-37Conference paper (Refereed)

This volume contains the proceedings of the conference on Manifolds, K-Theory, and Related Topics, held from June 23-27, 2014, in Dubrovnik, Croatia. The articles contained in this volume are a collection of research papers featuring recent advances in homotopy theory, $K$-theory, and their applications to manifolds. Topics covered include homotopy and manifold calculus, structured spectra, and their applications to group theory and the geometry of manifolds. This volume is a tribute to the influence of Tom Goodwillie in these fields.

4.

Arone, Gregory

et al.

Stockholm University, Faculty of Science, Department of Mathematics.

We study the equivariant homotopy type of the poset of orthogonal decompositions of a finite-dimensional complex vector space. Suppose that n is a power of a prime p, and that D is an elementary abelian p-subgroup of U(n) acting on complex n-space by the regular representation. We prove that the fixed point space of D acting on the decomposition poset of complex n-space contains as a retract the unreduced suspension of the Tits building for GL(k), which a wedge of (k-1)-dimensional spheres. Let Gamma be the projective elementary abelian subgroup of U(n) that contains the center of U(n) and acts irreducibly on complex n-space. We prove that the fixed point space of Gamma acting on the space of proper orthogonal decompositions of complex n-space is homeomorphic to a symplectic Tits building, which is also a wedge of (k-1)-dimensional spheres. As a consequence of these results, we find that the fixed point space of any coisotropic subgroup of Gamma contains, as a retract, a wedge of (k-1)-dimensional spheres. We make a conjecture about the full homotopy type of the fixed point space of D, based on a more general branching conjecture, and we show that the conjecture is consistent with our results.

Bredon homology of partition complexes2016In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 21, p. 1227-1268Article in journal (Refereed)

Abstract [en]

We prove that the Bredon homology or cohomology ofthe partition complex with fairly general coefficients is either trivialor computable in terms of constructions with the Steinberg module.The argument involves a theory of Bredon homology and cohomologyapproximation.

6.

Stålhammar, Marcus

et al.

Stockholm University, Faculty of Science, Department of Physics.

Rødland, Lukas

Arone, Gregory

Stockholm University, Faculty of Science, Department of Mathematics.

Budich, Jan Carl

Bergholtz, Emil J.

Stockholm University, Faculty of Science, Department of Physics.

We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum- and lattice models with relatively short-ranged hopping that is within experimental reach. To determine the topology of the nodal structures, we devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.