By Dominique Arlettaz

The second one Arolla convention on algebraic topology introduced jointly experts protecting quite a lot of homotopy thought and $K$-theory. those complaints replicate either the range of talks given on the convention and the variety of promising learn instructions in homotopy conception. The articles contained during this quantity contain major contributions to classical risky homotopy thought, version type thought, equivariant homotopy thought, and the homotopy concept of fusion structures, in addition to to $K$-theory of either neighborhood fields and $C^*$-algebras

Show description

Read or Download An Alpine Anthology of Homotopy Theory PDF

Similar algebraic geometry books

Geometric Integration Theory

This remedy of geometric integration conception involves an advent to classical concept, a postulational method of basic idea, and a bit on Lebesgue conception. Covers the idea of the Riemann vital; summary integration concept; a few family members among chains and services; Lipschitz mappings; chains and additive set capabilities, extra.

Lectures on Resolution of Singularities

Solution of singularities is a strong and regularly used instrument in algebraic geometry. during this ebook, J? nos Koll? r offers a entire therapy of the attribute zero case. He describes greater than a dozen proofs for curves, many in response to the unique papers of Newton, Riemann, and Noether. Koll?

Singularities in Algebraic and Analytic Geometry

This quantity includes the lawsuits of an AMS specified consultation held on the 1999 Joint arithmetic conferences in San Antonio. The members have been a global workforce of researchers learning singularities from algebraic and analytic viewpoints. The contributed papers include unique effects in addition to a few expository and ancient fabric.

Real and Complex Singularities: Ninth International Workshop on Real and Copmplex Singularities July 23-28, 2006 Icmc-usp, Sao Carlos, S.p., Brazil

This ebook bargains a range of papers in accordance with talks on the 9th overseas Workshop on genuine and intricate Singularities, a sequence of biennial workshops equipped via the Singularity idea team at Sao Carlos, S. P. , Brazil. The papers take care of the entire diverse subject matters in singularity conception and its functions, from natural singularity thought concerning commutative algebra and algebraic geometry to these issues linked to a number of points of geometry to homotopy concept

Additional resources for An Alpine Anthology of Homotopy Theory

Example text

4. 1 is complete. 2 (p. 5). If |b| < 1 and a is an arbitrary complex number, then ∞ ∞ (−1)n (−q; q)n (−aq/b; q)n bn (−1)n (−aq/b; q)n bn q n(n+1)/2 = . 2 (aq; q )n+1 (−b; q)n+1 n=0 n=0 Proof. 1), set h = 2 and t = q 2 , and replace b, c, and a by −b, aq, and aq, respectively. 1) with h = 1, q replaced by q 2 , t = q 2 , and a, b, and c replaced by −b, −bq, and aq 2 , respectively. 1) with q replaced by q 2 , α = −aq/b, β = −bq 2 , and τ = −bq to deduce that, after multiplying both sides by 1/(1 + b), ∞ (−aq/b; q 2 )n (−bq)n 2) (−b; q n+1 n=0 ∞ = (−aq/b; q 2 )n (−aq 2 /b; q 2 )n (−bq 2 )n (−bq)n q 2n (−b; q 2 )n+1 (−bq; q 2 )n+1 n=0 ∞ 2 −2n (1 − aq 4n+2 ) 2 (−aq/b; q 2 )n (−aq 2 /b; q 2 )n b2n q 2n +n (1 − aq 4n+2 ) = (−b; q 2 )n+1 (−bq; q 2 )n+1 n=0 = = ∞ 2 ∞ 2 (−aq/b; q)2n b2n q 2n +n (1 − aq 4n+2 ) (−b; q)2n+2 n=0 (−aq/b; q)2n b2n q 2n (−b; q)2n+1 n=0 ∞ (−aq/b; q)2n b2n q 2n = (−b; q)2n+1 n=0 +n 1+ 2 +n −bq 2n+1 − aq 4n+2 1 + bq 2n+1 ∞ − (−aq/b; q)2n+1 b2n+1 q (n+1)(2n+1) (−b; q)2n+2 n=0 ∞ = (−1)n (−aq/b; q)n bn q n(n+1)/2 , (−b; q)n+1 n=0 which is the desired result.

11). For any complex number a, ∞ (q 2 ; q 4 )∞ ∞ (aq 2 ; q 2 )n q n(n+1)/2 (aq 2 ; q 4 )n q 4n = (aq 4 ; q 4 )∞ (q; q)n (q 2 ; q 2 )2n n=0 n=0 2 ∞ + (aq 2 ; q 4 )∞ 2 (aq 4 ; q 4 )n q 4n +4n+1 . (q 2 ; q 2 )2n+1 n=0 Proof. 13, replace q by q 2 and set b = −1/q. This yields ∞ 1 an q 2n = 4 ; q 4 ) (−q; q 2 ) 2 ; q 4 ) (−q; q 2 ) (q (aq n n ∞ ∞ n=0 + ∞ (aq 2 ; q 4 )n q 4n (q 2 ; q 2 )2n n=0 1 4 4 (aq ; q )∞ (−q; q 2 )∞ ∞ 2 2 (aq 4 ; q 4 )n q 4n +4n+1 . 1). More precisely, let h = 2, c = −q, and a = 0, and let b tend to 0.

17) which is implicit in the work of Ramanujan in his lost notebook [244]. 2]. 10 involving two additional parameters. 10 (p. 10). ∞ 1 qn = 2 (q)n (q)2∞ n=0 ∞ (−1)n q n(n+1)/2 . 18) n=0 Proof. 1), set h = 1, t = c = q, and a = 0, and then let b → 0. 10 follows immediately. 11 (p. 10). ∞ ∞ 1 q 2n = 2 2 (q) (q) n ∞ n=0 (−1)n q n(n+1)/2 1+2 . 19) n=1 Proof. 1), set h = 1, a = 0, c = q, and t = q 2 . Now let b → 0 to deduce that ∞ ∞ n+1 1 q 2n n1 − q q n(n+1)/2 = (1 − q) (−1) 2 2 (q) (q) 1 − q n ∞ n=0 n=0 1 = (q)2∞ = 1 (q)2∞ ∞ ∞ n n(n+1)/2 (−1) q n=0 (−1)n+1 q (n+1)(n+2)/2 + n=0 ∞ (−1)n q n(n+1)/2 1+2 .

Download PDF sample

Download An Alpine Anthology of Homotopy Theory by Dominique Arlettaz PDF
Rated 4.18 of 5 – based on 32 votes