On homotopy nilpotency

We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces \(\Omega(G/K)\) of homogenous spaces \(G/K\) for a compact Lie group \(G\) and its closed homotopy nilpotent subgroup \(K \lt G\) is discussed.

On homotopy nilpotency of some suspended spaces

A homological criterium from [Golasiński, M., On homotopy nilpotency of loop spaces of Moore spaces, Canad. Math. Bull. (2021), 1–12] is applied to investigate the homotopy nilpotency of some suspended spaces. We investigate the homotopy nilpotency of the wedge sum and smash products of Moore spaces M (A, n) with n ≥ 1. The homotopy nilpotency of homological spheres are studied as well.

A construction of complex analytic elliptic cohomology from double free loop spaces

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$ , the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$ .

On the infinite loop spaces of algebraic cobordism and the motivic sphere

We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where $\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp. $\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$, $\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic. Comment: 13 pages. v5: published version; v4: final version, to appear in \'Epijournal G\'eom. Alg\'ebrique; v3: minor corrections; v2: added details in the moving lemma over finite fields

Homotopy nilpotency of some homogeneous spaces

Let $${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$ K = R , C , the field of reals or complex numbers and $${\mathbb {H}}$$ H , the skew $${\mathbb {R}}$$ R -algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$ Ω ( G n , m ( K ) ) , $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$ Ω ( F n ; n 1 , … , n k ( K ) ) , and $$\Omega (V_{n,m}({\mathbb {K}}))$$ Ω ( V n , m ( K ) ) of Grassmann $$G_{n,m}({\mathbb {K}})$$ G n , m ( K ) , flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$ F n ; n 1 , … , n k ( K ) and Stiefel $$V_{n,m}({\mathbb {K}})$$ V n , m ( K ) manifolds. Additionally, homotopy nilpotency classes of p-localized $$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( G n , m + ( K ) ( p ) ) and $$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( V n , m ( K ) ( p ) ) for certain primes p are estimated, where $$G^+_{n,m}({\mathbb {K}})_{(p)}$$ G n , m + ( K ) ( p ) is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.

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.