Abstract
We study the restrictions, the strict fixed points, and the strict quotients of the partition complex |Pi_{n}|, which is the Sigma_{n}-space attached to the poset of proper nontrivial partitions of the set {1,ldots,n}.We express the space of fixed points |Pi_{n}|^{G} in terms of subgroup posets for general Gsubset Sigma_{n} and prove a formula for the restriction of |Pi_{n}| to Young subgroups Sigma_{n_{1}}times cdotstimes Sigma_{n_{k}}. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of |Pi_{n}|, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients |Pi_{n}|^{diamond} mathbin {operatorname* {wedge }_{Sigma_{n}}^{}} (S^{ell})^{wedge n} and give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of |Pi_{n}| in terms of “atoms” |Pi_{d}|^{diamond} mathbin {operatorname* {wedge }_{Sigma_{d}}^{}} (S^{ell})^{wedge d} for ell odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from mathbf {F}_{2} to mathbf {F}_{p} for p an odd prime.
Highlights
We uncover surprising links between strict Young quotients of | n|, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry
We describe general fixed points, Young restrictions, and strict Young quotients of | n|
Example. — If we feed our algorithm 4 with its 2 × 2-action and a suitably chosen pair of functions, it collapses the contractible subcomplex drawn with thin lines on the left and gives rise to the bouquet of circles on the right: In general, our branching rule for restrictions of the partition complex n to Young subgroups n1 × · · · × nk reads
Summary
We shall begin by describing our general combinatorial branching algorithm and proceed to explain its consequences for fixed points and restrictions of partition complexes and Bruhat-Tits buildings. P n is isomorphic to the poset BT(Fkp) of proper non-trivial subgroups of P, which is closely related to the Tits building for GLk(Fp) Combining these observations with our Theorem 6.2 and Lemma 6.3, we can complete the calculation and describe all fixed point spaces of the partition complex with respect to p-groups: Corollary 6.8. We first give an explicit point-set level description of the map in (1.1) as a collapse map and prove that it is an equivariant simple homotopy equivalence of spaces. The preprint motivated the secondnamed author to strengthen his discrete Morse theoretic methods, develop the theory of orthogonality fans, and thereby give the direct combinatorial proof of the strengthened version of Theorem 5.10 presented in this paper In this approach, fixed points and restrictions are computed independently by applying a general combinatorial technique
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