Last time we discussed functoriality: how continuous maps between spaces (or pairs of spaces) help transfer homological (or relative homologi cal) information. Today we will see that these maps allow us to see connections between: • the relative homology of a pair of spaces (X,A) and the absolute homology of the two spaces X andA. • the homology of a decomposed space X = X ∪X in terms of the homology of the spacesX, X, andX ∩ X. The second idea in particular is very important, for it will a l ow us to understand the homology of complicated spaces via a glueing-together of si mpler spaces. For both ideas, the key concept is that all of the relevant homology gr oups fit together into a long exact sequence; we begin with an abstract description o f this algebraic concept, before moving on to more concrete examples.

In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it. Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution. The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states. 1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes xn = 0, xn = xn−1 = 0, etc... 2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group Gk would be analogous to the relative homology group relative to the plane xk−1 = 0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description. 3. The algebraic counterpart of boundary map would map the elements of Gk identified as analog of homotopy group to the commutator group [Gk−2, Gk−2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group

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Motives and Infinite Primes

. We ﬁx motivic data ( K/F, E ) consisting of a Galois extension K/F of characteristic p global ﬁelds with arbitrary abelian Galois group G and an ableian t –module E , deﬁned over a certain Dedekind subring of F . For this data, one can deﬁne a G –equivariant motivic L –function Θ EK/F . We reﬁne the techniques developed in [5] and prove an equivariant Tamagawa number formula for appropriate Euler product completions of the special value Θ EK/F (0) of this equivariant L –function. This extends our results in [5] from the Drinfeld module setting to the t –module setting. As a ﬁrst notable consequence, we prove a t –module analogue of the classical (number ﬁeld) Reﬁned Brumer–Stark Conjecture, relating a certain G –Fitting ideal of the t –motive analogue H ( E/K ) of Taelman’s class modules [9] to the special value Θ EK/F (0) in question. As a second consequence, we prove formulas for the values Θ EK/F ( m ), at all positive integers m ∈ Z ≥ 0 , when E is a Drinfeld module. This, in turn, implies a Drinfeld module analogue of the classical Reﬁned Coates–Sinnott Conjecture relating Θ EK/F ( m ) to the Fitting ideals of certain Carlitz twists H ( E ( m ) / O K ) of Taelman’s class modules, suggesting a strong analogy between these twists and the even Quillen K –groups of a number ﬁeld. In an upcoming paper, these consequences will be used to develop an Iwasawa theory for the t –module analogues of Taelman’s class modules.

An Equivariant Tamagawa Number Formula for t-Modules and Applications

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Motives and Infinite Primes

An Equivariant Tamagawa Number Formula for t-Modules and Applications