In this paper we introduce a general method of constructing stable operations which can be applied to any equivariant cohomology theory represented by a genuine equivariant highly commutative ring spectrum. We observe that our method constructs all Steenrod operations in the case when the group is either  trivial or have order two. Moreover, we construct new families of nontrivial cohomology operations for all finite groups. 

https://arxiv.org/abs/2410.12087

This erratum remedies errors in the literature pertaining to the stable Adams conjecture. As part of the above corrections, we also identify and fix two errors in section 4 of our recent article on the subject. We thank E. Fridelander for flagging these oversights and for offering helpful suggestions. This erratum is self-contained and also includes an appendix proving a version of Friedlander's classification result for sectioned fibrations of Gamma-spaces, after making appropriate changes to the original statement as indicated in the appendix.

https://arxiv.org/abs/2410.12087

Weiss  calculus of functor  is  a  tool  that facilitates  the use of  stable  calculations  to  explore  unstable  problems in  homotopy theory  which are generally  closer to  geometry.  In this paper we  lay down the  theoretical  foundations  of  equivariant  Weiss  calculus  which  may have  many potential applications to  equivariant geometry. 

https://arxiv.org/abs/2404.10062

In this paper, we  identify seven new 192-periodic infinite families in the stable homotopy groups of spheres which are nonzero after K(2)- as well as T(2)-localization.  

https://arxiv.org/pdf/2303.10259.pdf

In this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra (the cohomology theory formed out of classifying spaces of spherical bundles). This viewpoint allows us to identify, for a finite group G, a precise condition under which an R-orientation of a G-equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Steifel-Whitney class of a "homogeneous" G-equivariant bundle with respect to an commutative ring spectrum R. As an application, we show that the 2-fold direct sum of any homogeneous bundle is orientable with respect to Burnside ring. We notice that  orientability with respect to Burnside ring  is equivalent to orientability with respect to the ring of integers when the order of G is odd. When the order of G is even, we show that a G-equivariant analog of the tautological line bundle over the infinite projective space is orientable with respect to the integers, but not with respect to the Burnside ring. 

https://arxiv.org/abs/2309.16142

In this paper, we demonstrate how to calculate cohomology of the Spanier-Whitehead dual of a R-motivic finite spectra when its cohomology is free over the coefficient ring. We implement this method to deduce that only 16 out of 128 different R-motivic A(1) are Spanier-Whitehead self-dual.

https://arxiv.org/pdf/2301.11230.pdf

In this paper we calculate the E_1 page of the v_2-local as well as g-local algebraic tmf resolution. Here g is the element in the Ext over A(2) which detects the kappabar in the Hurewicz image of tmf. 

https://arxiv.org/abs/2106.10769

In this paper , we not only  realize  the  subalgebra A^R(1) of the R-motivic Steenrod algebra, we also achieve a other few important goals. Firstly, we provide a version of R-motivic Toda realization. Secondly, we draw a connection between the action of  RO(C_2)-squaring operations on the cohomology of a given C_2-equivariant space with the classical Steenrod operations on the cohomology of both its underlying as well as its geometric fixed-points. Finally, we construct the analogues of two important classical finite spectra, namely A(2) and Z.  

https://arxiv.org/abs/2008.05547

In this paper we consider a particular action of C_2 on the suspension spectrum of  RP^2 smash CP^2 and show it admits a C_2 equivariant  v_1-self-map of periodicity 1.  This is the first example of an equivariant v_1-self-map. In fact, we prove an R_motivic analogue of this result and obtain the C_2-equivariant result by applying the Betti realization functor.

https://arxiv.org/abs/2003.03795

We study the orientability of vector bundles with respect to a family of cohomology theories called EO-theories.  From chromatic point of view EO-theories are higher height analogue of real K-theory KO.  For each EO-theory we prove that there exists a fixed i such that for any vector bundle, its i fold direct sum is EO-orientable. 

https://arxiv.org/abs/1909.13379

In this paper, we study the tmf resolution of the spectrum Z which has the special property that it admits a v_2-self-map of periodicity 1. In particular, we prove a conjecture of https://arxiv.org/abs/1706.06170 which completes the calculation of the K(2)-local homotopy groups of Z-- the easier side of the telescope conjecture. Further, we speculate on how one could potentially resolve the telescope conjecture using the tmf resolution of Z. 

https://arxiv.org/abs/1810.05622

P_2^1 is a certain element of the Steenrod algebra such that its square is zero. Therefore its acts as a differential on the cohomology of a space  or a spectrum. The resultant homology groups are called P_2^1 Margolis homology.  We calculate the P_2^1 Margolis homology of the spectrum tmf and its smash powers. The real challenge lies in the fact that the action of P_2^1 does not follow liebniz rule. We expect these calculations to be a key component in the study of tmf-resolution of the sphere spectrum. 

https://arxiv.org/abs/1803.11014

In this paper, we provide a new proof of the stable Adams conjecture. Our proof constructs a canonical null-homotopy of the stable J-homomorphism composed with a virtual Adams operation, by applying the K-theory functor to a multi-natural transformation. We also point out that the original proof of the stable Adams conjecture is incorrect and present a correction. This correction is crucial to our main application. We settle the question on the height of higher associative structures on the mod p^k Moore spectrum M_p(k) at odd primes. More precisely, for any odd prime p, we show that M_p(k)admits a thomified A_n-structure if and only if  n<p^k. We also prove a weaker result for p=2.

https://arxiv.org/abs/1702.00230

The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the v_1-torsion part to the E_2-term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.

https://arxiv.org/abs/1702.01493 

A(n) is the sub-algebra of the generated by Sq^1, Sq^2, ..., Sq^{2^n}. The stable Picard group of A(n) consists of those finite A(n)-modules which are stably invertible under tensor product.  While the stable picard group of A(1) is Z/2, to our surprise it turns out that the stable picard group of A(2) is trivial. 

https://arxiv.org/abs/1706.06170

In this paper, we notice that the height 2  Morava E-homology  of the spectrum Z introduced in https://arxiv.org/abs/1608.0625 exhibits the regular representation of the quaternion group. This fact allows us to exploit the homotopy fixed point spectral sequence to calculate the K(2)-local homotopy groups of Z. Had it not been for the possibility of a couple of d_3-differentials, we would have completely calculated the K(2)-local homotopy groups of Z. We conjecture that the potential d_3-differentials are trivial.

https://arxiv.org/abs/1607.02702

The Moore spectrum M_p(i) is the cofiber of the p^i-map on the sphere spectrum.  In this paper we make use of the fact that the Moore spectrum is a Thom spectrum and a theorem of Stasheff to obtain a lower bound on i (for fixed p and n) for which M_p(i) is guaranteed to admit an A_n-structure. 

https://arxiv.org/abs/1608.06250

We introduce a class of type 2 spectrum which admit a 1-periodic v_2-self-map. Before this work it was unclear if there were any finite spectra with v_2-self-map of periodicity less than 32. 

https://arxiv.org/abs/1406.3297

Any spectrum whose cohomology realizes the subalgebra A(1) =< Sq^1, Sq^2 > of the Steenrod algebra is called A_1. This paper proves that A_1 admits a v_2-self-map of periodicity 32 which is minimal. 

https://arxiv.org/abs/1504.01408

We show that p-adic integers has properties like fractals, or in abstract language it is a final coalgebra to a certain functor. This universal property constructs both the addition and the multiplication as a final map. 

In this paper we generalize Freyd's example of unit interval as a final coalgebra to obtain Sierpinski's triangle with path metric as a final coalgebra.