Loday constructions on twisted products and on tori. (English) Zbl 1496.18017

Hedenlund, Alice; Klanderman, Sarah; Lindenstrauss, Ayelet; Richter, Birgit; Zou, Foling

Topology Appl. 316, Article ID 108103, 25 p. (2022).

“We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted Cartesian products in the case where the group involved is discrete. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Y. Berest et al. [Int. Math. Res. Not. 2022, No. 6, 4093–4180 (2022; Zbl 1494.57050)], but prove that several Loday constructions of truncated polynomial rings with reduced coefficients are not stable by investigating their torus homology.”
This extensive work has four sections. The first section is entitled “The Loday construction: basic features”. The authors recall some definition concerning Loday construction [J.-L. Loday, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)] and fix notation. The authors write instead “For most of our work we can use any good symmetric monoidal category of spectra whose category of commutative monoids is Quillen equivalent to the category of \(E_\infty\)-ring spectra, such as symmetric spectra [M. Hovey et al., J. Am. Math. Soc. 13, No. 1, 149–208 (2000; Zbl 0931.55006)], orthogonal spectra [M. A. Mandell and J. P. May, Equivariant orthogonal spectra and \(S\)-modules. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1025.55002)] or \(\textsl{\textbf{S}}\)-modules [A. D. Elmendorf et al., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Providence, RI: American Mathematical Society (1997; Zbl 0894.55001)]. As parts of the paper require us to work with a specific model category we chose to work with the category of \(S\)-modules everywhere except in Section 3, where we will work in the \(\infty\)-category of spectra in the sense of Luri [J. Lurie, Higher algebra. (2017), http://www.math.harvard.edu/-lurie/papers/HA.pdf].”
Let \(X\) be a finite pointed simplicial set and \(R\rightarrow A\rightarrow C\) be a sequence of maps of commutative ring spectra.
{Definition 1.1}. The Loday construction with respect to \(X\) of \(A\) over \(R\) with coefficients in \(C\) is the simplicial commutative augumented \(C\) -algebra spectrum \(\mathfrak{L}^R_X(A;C)\) given by \(\mathfrak{L}^R_X(A;C)_n=C\wedge \bigwedge_{x\in X_n\backslash \ast} A\), where the smash products are taken over \(R\). Here, \(\ast\) denotes the basepoint of \(X\) and we place a copy of \(C\) at the basepoint.
The authors assume in addition that \(R\) is a cofibrant commutative \(S\)-algebra, \(A\) is a cofibrant commutative \(R\)-algebra and \(C\) is a cofibrant commutative \(A\)-algebra. This ensures that the homotopy type of \(\mathfrak{L}^R_X(A;C)\) is well-defined and depends only on the homotopy type of \(X\). Under these conditions, new notations are adopted for some used in [J.-L. Loday, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007); A. D. Elmendorf et al., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Providence, RI: American Mathematical Society (1997; Zbl 0894.55001); T. Pirashvili, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 2, 151–179 (2000; Zbl 0957.18004)]. Thus \(\mathfrak{L}^R_X(A;C)\) instead of \(C\otimes\bigotimes_{x\in {X_n}\setminus\ast}A\), in [J.-L. Loday, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)], for \(R\rightarrow A\rightarrow C\) a sequence of commutative rings. And, for \(X=S^n\), \(\mathfrak{L}^R_{S^n}(A;C)\), instead of \(THH^{[n],R}(A;C)\), in [A. D. Elmendorf et al., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Providence, RI: American Mathematical Society (1997; Zbl 0894.55001)] and [T. Pirashvili, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 2, 151–179 (2000; Zbl 0957.18004)].
Section 2, entitled “A spectral sequence for twisted Cartesian products”, contains the construction of a spectral sequence \(E^2_{p,q}=\pi_p((\mathfrak{L}^R_B(\pi_\ast \mathfrak{L}^R_F(A)^\tau))_q)\Rightarrow \pi_{p+q}(\mathfrak{L}^R_{E(\tau)}(A))\) (Theorem 2.10) for the homotopy groups \(\pi_\ast(\mathfrak{L}^R_F(A))\) of Loday constructions with respect to a twisted Cartesian products (TCP), \(E(\tau)=F\times_\tau B, \) [J. P. May, Simplicial objects in algebraic topology. Chicago: The University of Chicago Press (1992; Zbl 0769.55001)]. Two examples are given.One starting from a TCP constructed by the connected \(n\)-fold of \(S^1\) given by degree \(n\) map. And the second,conversely, for the Klein bottle, \(K\ell\), is recomputed the homotopy groups of the Loday construction of the polynomial algebra \(k[x]\) for a field \(k\), such that 2 is invertible in \(k\), using the above TCP spectral sequence.
Section 3, “Hopf algebras in spectra”, the authors prove that the Loday construction is stable for commutative Hopf algebra spectra, generalizing a result of Y. Berest et al. [Int. Math. Res. Not. 2022, No. 6, 4093–4180 (2022; Zbl 1494.57050)]. If CAlg denote the \(\infty\)-category of \(E_\infty\)-ring spectra, then a commutative Hopf algebra spectrum is a cogroup in CAlg. In the introduction of this section the authors give some examples of Hopf algebra spectra. A first example starts from a topological abelian group \(G\), with the spherical group ring \(S[G]=\sum_{+}^\infty G\), equipped with the product induced by the product in \(G\), the coproduct induced by the diagonal map \(G\rightarrow G\times G\), and the antipodal map induced by the inverse map from \(G\) to \(G\) is a commutative Hopf algebra spectrum. Another example starts from an ordinary commutative Hopf algebra \(A\) over a commutative ring \(k\) and A is flat as a \(k\)-module. Then the Eilenberg -Mac Lane spectrum \(HA\) is a commutative Hopf algebra spectrum over \(Hk\). Other examples are also given and then is proved the following theorem.
{Theorem 3.6}. If \(\mathcal{H}\) is a commutative Hopf algebra spectrum and if \(\sum(X_+)\simeq \sum(X_+)\) is an equivalence in \(\mathcal{S}_\ast\) , then there is an equivalence \(X\otimes \mathcal{H}\simeq Y\otimes \mathcal{H}\) in CAlg. (Where \(\mathcal{S}_\ast\) denotes the \(\infty\)-category of based spaces).
Section 4 is entitled “Truncated polynomial algbras”. But the notations in this part of the article are too complicated to be transcribed or summarized by the reviewer even only in the two big theorems (4.10 and 4.23) in this section. That is why the reviewer is limited to what the authors wrote in relation to this section. “In Section 4 we prove that truncated polynomial algebras of the form \(\mathbb{Q}[t]/t^m\) and \(\mathbb{Z}[t]/t^m\) for \(m\geq 2\) are not muliplicatively stable by comparing the Loday construction of tori to the Loday construction of a bouquet of spheres corresponding to the cells of the tory. We also show that for \(2\leq m< p\) the pairs \((\mathbb{F}_p[t]/t^m;\mathbb{F}_p)\) are not stable”.

Reviewer: Ioan Pop (Iaşi)

MSC:

18G90	Other (co)homology theories (category-theoretic aspects)
55P43	Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)

Keywords:

torus homology; (higher) Hochschild homology; (higher) topological Hochschild homology; stability; twisted cartesian products

Citations:

Zbl 0885.18007; Zbl 0931.55006; Zbl 1025.55002; Zbl 0894.55001; Zbl 0957.18004; Zbl 0769.55001; Zbl 1494.57050

References:

[1]	Adamyk, Katharine; Gerhardt, Teena; Hess, Kathryn; Klang, Inbar; Kong, Hana Jia, Computational tools for twisted topological Hochschild homology of equivariant spectra, Topol. Appl., 316, Article 108102 pp. (2022) · Zbl 1493.55015
[2]	Angeltveit, Vigleik; Blumberg, Andrew J.; Gerhardt, Teena; Hill, Michael A.; Lawson, Tyler; Mandell, Michael A., Topological cyclic homology via the norm, Doc. Math., 23, 2101-2163 (2018) · Zbl 1417.55015
[3]	Berest, Yuri; Ramadoss, Ajay C.; Yeung, Wai-Kit, Representation homology of topological spaces, Int. Math. Res. Not., 2022, 6, 4093-4180 (2022) · Zbl 1494.57050
[4]	Bobkova, Irina; Höning, Eva; Lindenstrauss, Ayelet; Poirier, Kate; Richter, Birgit; Zakharevich, Inna, Splittings and calculational techniques for higher \(\mathsf{THH} \), Algebraic Geom. Topol., 19, 7, 3711-3753 (2019) · Zbl 1436.55017
[5]	Bobkova, Irina; Lindenstrauss, Ayelet; Poirier, Kate; Richter, Birgit; Zakharevich, Inna, On the higher topological Hochschild homology of \(\mathbb{F}_p\) and commutative \(\mathbb{F}_p\)-group algebras, (Women in Topology: Collaborations in Homotopy Theory. Women in Topology: Collaborations in Homotopy Theory, Contemporary Mathematics, vol. 641 (2015), AMS), 97-122 · Zbl 1346.18026
[6]	Dundas, Bjørn Ian; Tenti, Andrea, Higher Hochschild homology is not a stable invariant, Math. Z., 290, 1-2, 145-154 (2018) · Zbl 1434.55005
[7]	Elmendorf, Anthony D.; Kriz, Igor; Mandell, Michael A.; May, J. Peter, Rings, Modules, and Algebras in Stable Homotopy Theory, Mathematical Surveys and Monographs, vol. 47 (1997), American Mathematical Society: American Mathematical Society Providence, RI, xii+249 · Zbl 0894.55001
[8]	Goerss, Paul G.; Jardine, John F., Simplicial Homotopy Theory, Modern Birkhäuser Classics (2009), Birkhäuser Verlag, xv+510 pp. · Zbl 1195.55001
[9]	Hovey, Mark; Shipley, Brooke; Smith, Jeff, Symmetric spectra, J. Am. Math. Soc., 13, 1, 149-208 (2000) · Zbl 0931.55006
[10]	Lindenstrauss, Ayelet; Richter, Birgit, Stability of Loday constructions, Homol. Homotopy Appl., 24, 1, 401-425 (2022) · Zbl 1487.18014
[11]	Loday, Jean-Louis, Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, vol. 301 (1998), Springer-Verlag: Springer-Verlag Berlin, xx+513 pp. · Zbl 0885.18007
[12]	Lurie, Jacob, Higher Topos Theory, Annals of Mathematics Studies, vol. 170 (2009) · Zbl 1175.18001
[13]	Lurie, Jacob, Higher algebra (September 2017)
[14]	Mandell, Michael A.; May, J. Peter, Equivariant orthogonal spectra and S-modules, Mem. Am. Math. Soc., 159, 755 (2002), x+108 pp. · Zbl 1025.55002
[15]	May, J. Peter, Simplicial Objects in Algebraic Topology, Vol. 11 (1992), University of Chicago Press · Zbl 0769.55001
[16]	May, J. Peter, The Geometry of Iterated Loop Spaces, Lectures Notes in Mathematics, vol. 271 (1972), Springer-Verlag: Springer-Verlag Berlin-New York, viii+175 pp. · Zbl 0244.55009
[17]	Nikolaus, Thomas, The group completion theorem via localizations of ring spectra, expository notes, available at
[18]	Péroux, Maximilien; Shipley, Brooke, Coalgebras in symmetric monoidal categories of spectra, Homol. Homotopy Appl., 21, 1, 1-18 (2019) · Zbl 1433.16040
[19]	Pirashvili, Teimuraz, Hodge decomposition for higher order Hochschild homology, Ann. Sci. Éc. Norm. Supér. (4), 33, 2, 151-179 (2000) · Zbl 0957.18004
[20]	Rasekh, Nima; Stonek, Bruno; Valenzuela, Gabriel, Thom spectra, higher \(\mathsf{THH}\) and tensors in ∞-categories, Algebraic Geom. Topol. (2022), in press · Zbl 1506.55008
[21]	Schaper, Christoph, Suspensions of affine arrangements, Math. Ann., 309, 3, 463-473 (1997) · Zbl 0896.55008
[22]	Schlichtkrull, Christian, Higher topological Hochschild homology of Thom spectra, J. Topol., 4, 1, 161-189 (2011) · Zbl 1228.55006
[23]	Segal, Graeme, Configuration-spaces and iterated loop-spaces, Invent. Math., 21, 213-221 (1973) · Zbl 0267.55020