scholarly journals Strictly Commutative Realizations of Diagrams Over the Steenrod Algebra and Topological Modular Forms at the Prime 2

2013 ◽  
Vol 2014 (10) ◽  
pp. 2773-2813 ◽  
Author(s):  
Tyler Lawson ◽  
Niko Naumann
Keyword(s):  
Modular Forms ◽  
Steenrod Algebra ◽  
Topological Modular Forms ◽  
Prime 2

scholarly journals The Picard group of topological modular forms via descent theory

2016 ◽  
Vol 20 (6) ◽  
pp. 3133-3217 ◽  
Author(s):  
Akhil Mathew ◽  
Vesna Stojanoska
Keyword(s):  
Modular Forms ◽  
Picard Group ◽  
Descent Theory ◽  
Topological Modular Forms

scholarly journals Topological Modular Forms of Level 3

2009 ◽  
Vol 5 (2) ◽  
pp. 853-872 ◽  
Author(s):  
Mark Mahowald ◽  
Charles Rezk
Keyword(s):  
Modular Forms ◽  
Topological Modular Forms ◽  
Level 3

scholarly journals On the ring of cooperations for 2‐primary connective topological modular forms

2019 ◽  
Vol 12 (2) ◽  
pp. 577-657 ◽  
Author(s):  
M. Behrens ◽  
K. Ormsby ◽  
N. Stapleton ◽  
V. Stojanoska
Keyword(s):  
Modular Forms ◽  
Topological Modular Forms

scholarly journals Secondary invariants for string bordism and topological modular forms

2014 ◽  
Vol 138 (8) ◽  
pp. 912-970 ◽  
Author(s):  
Ulrich Bunke ◽  
Niko Naumann
Keyword(s):  
Modular Forms ◽  
Topological Modular Forms

scholarly journals 4-manifolds and topological modular forms

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Sergei Gukov ◽  
Du Pei ◽  
Pavel Putrov ◽  
Cumrun Vafa
Keyword(s):  
Modular Forms ◽  
Flavor Symmetry ◽  
Basic Properties ◽  
Topological Modular Forms ◽  
Witten Genus

Abstract We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0, 1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on ’t Hooft anomalies of 6d (1, 0) theories and a better understanding of the relation between 2d (0, 1) theories and TMF spectra.

scholarly journals Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application to Singer's cohomological transfer

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Ext Groups ◽  
Hit Problem ◽  
Prime 2

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This problem is the content of "hit problem" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the Adams $E_2$-term, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and the representation of the fourth transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. These new results confirmed Singer's conjecture for the monomorphism of the rank $4$ transfer. Our approach is different from that of Singer.

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