Motivic Classifying Spaces

2019 ◽  
pp. 253-276
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Symmetric Product ◽  
Homotopy Groups ◽  
Classifying Spaces ◽  
Connected Space ◽  
Cohomology Operation ◽  
Kunneth Formula ◽  
Cohomology Operations

This chapter focuses on motivic classifying spaces. It first connects the motives 𝑆∞ tr(𝕃𝑛) to cohomology operations on 𝐻2𝑛, 𝑛, at least when char(𝑘)=0. This parallels the Dold–Thom theorem in topology, which identifies the reduced homology ̃𝐻*(𝑋, ℤ) of a connected space 𝑋 with the homotopy groups of the infinite symmetric product 𝑆∞𝑋. A similar analysis shows that 𝔾𝑚 represents 𝐻1,1(−, ℤ), which allows us to describe operations on 𝐻1,1. The chapter then introduces the notion of scalar weight operations on 𝐻2𝑛, 𝑛. Afterward, it develops formulas for 𝑆𝓁tr(𝕃𝑛). These formulas imply that 𝑆∞ tr(𝕃𝑛) is a proper Tate motive, so there is a Künneth formula for them. The chapter culminates in a theorem demonstrating that β‎𝑃𝑏 is the unique cohomology operation of scalar weight 0 in its bidegree.

scholarly journals A Conjecture on Homotopy Groups of Spheres, Details on the Algebra of Higher Cohomology Operations

2009 ◽  
Vol 16 (1) ◽  
pp. 1-12
Author(s):  
Hans-Joachim Baues
Keyword(s):  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Cohomology Operations

Abstract The computation of the algebra of secondary cohomology operations in [Baues, The algebra of secondary cohomology operations, Birkhäuser Verlag, 2006] leads to a conjecture concerning the algebra of higher cohomology operations in general and an Ext-formula for the homotopy groups of spheres. This conjecture is discussed in detail in this paper.

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions II

1992 ◽  
Vol 122 (1-2) ◽  
pp. 127-135 ◽  
Author(s):  
John W. Rutter
Keyword(s):  
Group Extension ◽  
Equivalence Classes ◽  
Homotopy Groups ◽  
Geometric Proof ◽  
Homotopy Equivalence ◽  
Connected Space ◽  
Abelian Kernel ◽  
Simply Connected ◽  
Extension Sequence ◽  
Connected Spaces

SynopsisWe give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.

On the differentials in the Adams spectral sequence

1964 ◽  
Vol 60 (3) ◽  
pp. 409-420 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Higher Order ◽  
Adams Spectral Sequence ◽  
Steenrod Algebra ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Cohomology Operations ◽  
Image Position

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements ofcorresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

On the homotopy theory of sheaves of simplicial groupoids

1996 ◽  
Vol 120 (2) ◽  
pp. 263-290 ◽  
Author(s):  
André Joyal ◽  
Myles Tierney
Keyword(s):  
Homotopy Theory ◽  
Characteristic Classes ◽  
Spectral Sequences ◽  
Classifying Spaces ◽  
Sheaf Cohomology ◽  
Natural Context ◽  
Simplicial Sheaves ◽  
Cohomology Operations

The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]

Cohomology operations and duality

1968 ◽  
Vol 64 (1) ◽  
pp. 15-30 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Vector Spaces ◽  
Kronecker Products ◽  
Dual Vector ◽  
Alexander Duality ◽  
Cohomology Operation ◽  
Yield Information ◽  
Stable Cohomology ◽  
Steenrod Square ◽  
Cohomology Operations ◽  
Image Position

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operationwhich is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker productsif h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that

Homotopy colimits in the category of small categories

1979 ◽  
Vol 85 (1) ◽  
pp. 91-109 ◽  
Author(s):  
R. W. Thomason
Keyword(s):  
Geometric Topology ◽  
Homotopy Groups ◽  
Homotopy Equivalence ◽  
Classifying Spaces

In (13), Quillen defines a higher algebraic K-theory by taking homotopy groups of the classifying spaces of certain categories. Certain questions in K-theory then become questions such as when do functors induce a homotopy equivalence of classifying spaces, or when is a square of categories homotopy cartesian? Quillen has given some techniques for answering such questions. F. Waldhausen has extended these ideas in (19), and broadened the range of applications to include geometric topology (20).

scholarly journals Classification of Mappings of an (n + 2)-Complex into an (n − 1)-Connected Space with Vanishing (n + 1)-st Homotopy Group

1952 ◽  
Vol 4 ◽  
pp. 43-50 ◽  
Author(s):  
Nobuo Shimada ◽  
Hiroshi Uehara
Keyword(s):  
Finite Number ◽  
Homotopy Group ◽  
Extension Theorem ◽  
Homotopy Groups ◽  
Connected Space ◽  
Number Of Generators

The present paper is concerned with the classification and corresponding extension theorem of mappings of the (n+-2)-complex Kn+2 (n>2) into the space Y whose homotopy groups πi(Y) vanish for i < n and i = n+1, and the n-th homotopy group πn(Y) of which has a finite number of generators. Our methods followed here are essentially analogous to those of Steenrod [2].

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