Subgroups of Finite Index in Profinite Groups
... Proof. Suppose that H and K are normal subgroups of G with G/H and G/K belonging to N l−1 . We will show by induction on l that G/(H ∩ K) belongs to N l−1 . If l = 1, then G is nilpotent, and therefore G/(H ∩ K) is a quotient of a nilpotent group. As factor groups of nilpotent groups are nilpotent, ...
... Proof. Suppose that H and K are normal subgroups of G with G/H and G/K belonging to N l−1 . We will show by induction on l that G/(H ∩ K) belongs to N l−1 . If l = 1, then G is nilpotent, and therefore G/(H ∩ K) is a quotient of a nilpotent group. As factor groups of nilpotent groups are nilpotent, ...
The Type of the Classifying Space of a Topological Group for the
... In Section 1 we will explain this notion and put it into context with the similar notion due to tom Dieck [4, section I.6]. We mention that these spaces E(G, F ) and in particular EG play an important role in the formulation of the Baum-Connes Conjecture [3, Conjecture 3.15 on p. 254], the Isomorphi ...
... In Section 1 we will explain this notion and put it into context with the similar notion due to tom Dieck [4, section I.6]. We mention that these spaces E(G, F ) and in particular EG play an important role in the formulation of the Baum-Connes Conjecture [3, Conjecture 3.15 on p. 254], the Isomorphi ...
introduction to banach algebras and the gelfand
... and by using their properties he created the modern theory of BA. Later (1941) these results are found in [Gelfand 1941]. In [Gelfand, Naimark 1943] Gelfand and Naimark3 proved the two major representation theorems, named after them, which form the main body of the theory of BA. Mazur’s 1938 theorem ...
... and by using their properties he created the modern theory of BA. Later (1941) these results are found in [Gelfand 1941]. In [Gelfand, Naimark 1943] Gelfand and Naimark3 proved the two major representation theorems, named after them, which form the main body of the theory of BA. Mazur’s 1938 theorem ...
![arXiv:math/0607274v2 [math.GT] 21 Jun 2007](http://s1.studyres.com/store/data/004624723_1-689e7c37d19114c6d0cb2b7e64bf5915-300x300.png)
arXiv:math/0607274v2 [math.GT] 21 Jun 2007
... Abstract We study the topology of the boundary manifold of a line arrangement in CP2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial ∆(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arb ...
... Abstract We study the topology of the boundary manifold of a line arrangement in CP2 , with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial ∆(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arb ...
Integer Exponents
... Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = ...
... Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = ...
Elliptic Curves Lecture Notes
... algebraically closed and may have positive characteristic). Definition 1.1. An elliptic curve over k is a nonsingular projective algebraic curve E of genus 1 over k with a chosen base point O ∈ E. Remark. There is a somewhat subtle point here concerning what is meant by a point of a curve over a non ...
... algebraically closed and may have positive characteristic). Definition 1.1. An elliptic curve over k is a nonsingular projective algebraic curve E of genus 1 over k with a chosen base point O ∈ E. Remark. There is a somewhat subtle point here concerning what is meant by a point of a curve over a non ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.