RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

Freudenthal's theorem and spherical classes in H*QS0

This note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z}, \mathbb{Z}/p$, with a special focus on the case of p=2 related to the Curtis conjecture. We apply Freudenthal's theorem to prove a vanishing result for the unstable Hurewicz image of elements in ${\pi _*^s}$ that factor through certain finite spectra. After either p-localization or p-completion, this immediately implies that elements of well-known infinite families in ${_p\pi _*^s}$, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism ${_p\pi _*^s}\simeq {_p\pi _*}QS^0\to H_*(QS^0;\mathbb{Z} /p)$. We also observe that the image of the submodule of decomposable elements under the integral unstable Hurewicz homomorphism $\pi _*^s\simeq \pi _*QS^0\to H_*(QS^0;\mathbb{Z} )$ is given by $\mathbb{Z} \{h(\eta ^2),h(\nu ^2),h(\sigma ^2)\}$. We apply the latter to completely determine spherical classes in $H_*(\Omega ^dS^{n+d};\mathbb{Z} /2)$ for certain values of n>0 and d>0; this verifies Eccles' conjecture on spherical classes in $H_*QS^n$, n>0, on finite loop spaces associated with spheres.

Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

After recent work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem [M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. Math. (2), 184 (2016), 1–262], as well as Adams’ solution of the Hopf invariant one problem [J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (2), 72 (1960), 20–104], an immediate consequence of Curtis conjecture is that the set of spherical classes in H∗Q0S0 is finite. Similarly, Eccles conjecture, when specialized to X=Sn with n> 0, together with Adams’ Hopf invariant one theorem, implies that the set of spherical classes in H∗QSn is finite. We prove a filtered version of the above finiteness properties. We show that if X is an arbitrary CW-complex of finite type such that for some n, HiX≃0 for any i>n, then the image of the composition π∗ΩlΣl+2X→π∗QΣ2X→H∗QΣ2X is finite; the finiteness remains valid if we formally replace X with S−1. As an application, we provide a lower bound on the dimension of the sphere of origin on the potential classes of π∗QSn which are detected by homology. We derive a filtered finiteness property for the image of certain transfer maps ΣdimgBG+→QS0 in homology. As an application to bordism theory, we show that for any codimension k framed immersion f:M↬ℝn+k which extends to an embedding M→ℝd×ℝn+k, if n is very large with respect to d and k then the manifold M as well as its self-intersection manifolds are boundaries. Some results of this paper extend results of Hadi [Spherical classes in some finite loop spaces of spheres. Topol. Appl., 224 (2017), 1–18] and offer corrections to some minor computational mistakes, hence providing corrected upper bounds on the dimension of spherical classes H∗ΩlSn+l. All of our results are obtained at the prime p = 2.

Exact Lorentz-violating q-deformed O(N) universality class

We examine the influence of exact Lorentz-violating symmetry mechanism on the radiative quantum corrections to the critical exponents for massless [Formula: see text]-deformed O([Formula: see text]) [Formula: see text] scalar field theories. For that, we employ three different and independent field-theoretic renormalization group methods for computing analytically the [Formula: see text]-deformed critical exponents up to next-to-leading order. Then, we generalize the former finite loop level results for any loop order. We show that the Lorentz-violating [Formula: see text]-deformed critical exponents, obtained through the three methods, turn out to be identical and furthermore the same as their Lorentz-invariant [Formula: see text]-deformed ones. We argue that this result is in accordance with the universality hypothesis.

Unitary embeddings of finite loop spaces

In this paper we construct faithful representations of saturated fusion systems over discrete p-toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p-local compact groups. These conditions hold for the Clark–Ewing and Aguadé–Zabrodsky p-compact groups. We also show the existence of unitary embeddings of finite loop spaces.

Units in finite loop algebras of RA2 loops

Let [Formula: see text] be the loop algebra of a loop [Formula: see text] over a field [Formula: see text]. In this paper, we characterize the structure of the unit loop of [Formula: see text] modulo its Jacobson radical when [Formula: see text] is an [Formula: see text] loop obtained from the dihedral group of order [Formula: see text], [Formula: see text] is an odd number and [Formula: see text] is a finite field of characteristic [Formula: see text]. The structure of [Formula: see text] is also determined.