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1. Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

   https://doi.org/10.1353/ajm.2024.a928325  

   Kuhn, Nicholas J; Lloyd, Christopher_J R (June 2024, American Journal of Mathematics)

   abstract: In the early 1940s, P. A. Smith showed that if a finite $$p$$-group $$G$$ acts on a finite dimensional complex $$X$$ that is mod $$p$$ acyclic, then its space of fixed points, $X^G$, will also be mod $$p$$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $$G$$-space $$X$$ is a equivariant homotopy retract of the $$p$$-localization of a based finite $$G$$-C.W. complex, given $H 

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2. On the slice spectral sequence for quotients of norms of Real bordism

   https://doi.org/10.1112/topo.70015  

   Beaudry, Agnès; Hill, Michael_A; Lawson, Tyler; Shi, XiaoLin_Danny; Zeng, Mingcong (February 2025, Journal of Topology)

   Abstract In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm by permutation summands. These quotients are of interest because of their close relationship with higher real ‐theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories . These spectra serve as natural equivariant generalizations of connective integral Morava ‐theories. We provide a complete computation of the ‐localized slice spectral sequence of , where is the real sign representation of . To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the ‐based Adams spectral sequence in the category of ‐modules. Furthermore, we provide a full computation of the ‐localized slice spectral sequence of the height‐4 theory . The ‐slice spectral sequence can be entirely recovered from this computation. 

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3. Generalized $$\mathbb Z$$-homotopy fixed points of $$C_n$$ spectra with applications to norms of $$MU_{\mathbb R}$$

   Hill, Michael A; Zeng, Mingcong (January 2020, New York journal of mathematics)

   null (Ed.)

   We introduce a computationally tractable way to describe the $$\mathbb Z$$-homotopy fixed points of a $$C_{n}$$-spectrum $$E$$, producing a genuine $$C_{n}$$ spectrum $$E^{hn\mathbb Z}$$ whose fixed and homotopy fixed points agree and are the $$\mathbb Z$$-homotopy fixed points of $$E$$. These form the bottom piece of a contravariant functor from the divisor poset of $$n$$ to genuine $$C_{n}$$-spectra, and when $$E$$ is an $$N_{\infty}$$-ring spectrum, this functor lifts to a functor of $$N_{\infty}$$-ring spectra. For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $$E^{hn\mathbb Z}$$, giving the homotopy groups of the $$\mathbb Z$$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $$\mathbb Z$$-homotopy fixed point case, giving us a family of new tools to simplify slice computations. 

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4. Cyclic pairings and derived Poisson structures.

   Ramadoss, Ajay C; Zhang, Yining (April 2019, New York journal of mathematics)

   null (Ed.)

   By a fundamental theorem of D. Quillen, there is a natural duality - an instance of general Koszul duality - between differential graded (DG) Lie algebras and DG cocommutative coalgebras defined over a field k of characteristic 0. A cyclic pairing (i.e., an inner product satisfying a natural cyclicity condition) on the cocommutative coalgebra gives rise to an interesting structure on the universal enveloping algebra Ua of the Koszul dual Lie algebra a called the derived Poisson bracket. Interesting special cases of the derived Poisson bracket include the Chas-Sullivan bracket on string topology. We study the derived Poisson brackets on universal enveloping algebras Ua, and their relation to the classical Poisson brackets on the derived moduli spaces DRep_g(a) of representations of a in a finite dimensional reductive Lie algebra g. More specifically, we show that certain derived character maps of a intertwine the derived Poisson bracket with the classical Poisson structure on the representation homology HR(a, g) related to DRep_g(a). 

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5. The Chow t-structure on the ∞-category of motivic spectra

   https://doi.org/10.4007/annals.2022.195.25  

   Kong, Hana J.; Wang, Guozhen; Xu, Zhouli (February 2022, Annals of mathematics)

   We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k. 

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