scholarly journals Integration of the Kenzo System within SageMath for New Algebraic Topology Computations

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 722
Author(s):  
Julián Cuevas-Rozo ◽  
Jose Divasón ◽  
Miguel Marco-Buzunáriz ◽  
Ana Romero
Keyword(s):  
Computer Algebra ◽  
Algebraic Topology ◽  
Homotopy Groups ◽  
Spectral Sequences ◽  
Simplicial Objects

This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before.

Homotopy Groups and CW Complexes

2020 ◽  
pp. 11-20
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Space ◽  
Algebraic Topology ◽  
Weak Topology ◽  
Homotopy Theory ◽  
Fiber Bundles ◽  
Homotopy Groups ◽  
Cw Complex ◽  
Continuous Maps ◽  
Cw Complexes ◽  
Set Of Points

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

scholarly journals On Relative Homotopy Groups of Modules

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
C. Joanna Su
Keyword(s):  
Algebraic Topology ◽  
Homotopy Theory ◽  
Homotopy Groups ◽  
Module Theory

In his book “Homotopy Theory and Duality,” Peter Hilton described the concepts of relative homotopy theory in module theory. We study in this paper the possibility of parallel concepts of fibration and cofibration in module theory, analogous to the existing theorems in algebraic topology. First, we discover that one can study relative homotopy groups, of modules, from a viewpoint which is closer to that of (absolute) homotopy groups. Then, through the study of various cases, we learn that the classic fibration/cofibration relation does not come automatically. Nonetheless, the ability to see the relative homotopy groups as absolute homotopy groups, in a stronger sense, promises to justify our ultimate search.

Simplicial Objects in Algebraic Topology

1995 ◽  
Vol 79 (484) ◽  
pp. 242
Author(s):  
Ulrike Tillmann ◽  
P. May
Keyword(s):  
Algebraic Topology ◽  
Simplicial Objects

scholarly journals On the lambda algebra and Singer's cohomological transfer

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Algebraic Topology ◽  
Short Note ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Topological Spaces ◽  
Homotopy Groups ◽  
Divided Power ◽  
Efficient Tool ◽  
Stable Homotopy Groups ◽  
Prime 2

Write $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an important problem of Algebraic topology, because it is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres. A relatively efficient tool to describe this cohomology is the Singer algebraic transfer of rank $n$ in \cite{Singer}, which passes from a certain subquotient of a divided power algebra to the cohomology of $\mathbb A.$ Singer predicted that this transfer is a monomorphism, but this remains open for $n\geq 4.$ This short note is to verify the conjecture in the ranks 4 and 5 and some generic degrees.

scholarly journals The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime

2012 ◽  
Vol 207 ◽  
pp. 139-157
Author(s):  
Ryo Kato ◽  
Katsumi Shimomura
Keyword(s):  
Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Spectral Sequences ◽  
First Line ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Ext Groups ◽  
Bockstein Spectral Sequence

AbstractThe chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term of the spectral sequence is an Ext group of BP*BP-comodules. There is a sequence of Ext groups for nonnegative integers n with and there are Bockstein spectral sequences computing a module (n – s) from So far, a small number of the E1-terms are determined. Here, we determine the for p > 2 and n > 3 by computing the Bockstein spectral sequence with E1-term for s = 1, 2. As an application, we study the nontriviality of the action of α1 and β1 in the homotopy groups of the second Smith-Toda spectrum V(2).

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