《代數配邊理論》是一部很難得的介紹代數配邊理論的專著，內容精煉簡短。書中在講述了Quillen復配邊方法後，接著在固定域的光滑變數範疇上引進有向上同調理論的觀點，證明了這樣一個理論—范的存在性叫做代數配邊。書中也包括了一些計算和套用案例。讀者對象：數學專業的研究生和科研人員。
基本介紹
- 書名：代數配邊理論
- 作者：M.萊文 (Mark Levine)
- 出版日期：2013年11月1日
- 語種：英語, 簡體中文
- ISBN：9787510070297
- 品牌：世界圖書出版公司北京公司
- 外文名：Algebraic Cobordism
- 出版社：世界圖書出版公司北京公司
- 頁數：244頁
- 開本：24
- 定價：49.00
基本介紹,內容簡介,作者簡介,圖書目錄,文摘,
基本介紹
內容簡介
《代數配邊理論》由世界圖書出版公司北京公司出版。
作者簡介
作者：（美國）M.萊文（Mark Levine）
圖書目錄
1 Cobordism and oriented cohomology
1.1 Oriented cohomology theories
1.2 Algebraic cobordism
1.3 Relations with complex cobordism
2 The definition of algebraic cobordism
2.1 Oriented Borel-Moore functors
2.2 Oriented functors of geometric type
2.3 Some elementary properties
2.4 The construction of algebraic cobordism
2.5 Some computations in algebraic cobordism
3 Fundamental properties of algebraic cobordism
3.1 Divisor classes
3.2 Localization
3.3 Transversality
3.4 Homotopy invariance
3.5 The projective bundle formula
3.6 The extended homotopy property
4 Algebraic cobordism and the Lazard ring
4.1 Weak homology and Chern classes
4.2 Algebraic cobordism and K-theory
4.3 The cobordism ring of a point
4.4 Degree formulas
4.5 Comparison with the Chow groups
5 Oriented Borel-Moore homology
5.1 Oriented Borel-Moore homology theories
5.2 Other oriented theories
1.1 Oriented cohomology theories
1.2 Algebraic cobordism
1.3 Relations with complex cobordism
2 The definition of algebraic cobordism
2.1 Oriented Borel-Moore functors
2.2 Oriented functors of geometric type
2.3 Some elementary properties
2.4 The construction of algebraic cobordism
2.5 Some computations in algebraic cobordism
3 Fundamental properties of algebraic cobordism
3.1 Divisor classes
3.2 Localization
3.3 Transversality
3.4 Homotopy invariance
3.5 The projective bundle formula
3.6 The extended homotopy property
4 Algebraic cobordism and the Lazard ring
4.1 Weak homology and Chern classes
4.2 Algebraic cobordism and K-theory
4.3 The cobordism ring of a point
4.4 Degree formulas
4.5 Comparison with the Chow groups
5 Oriented Borel-Moore homology
5.1 Oriented Borel-Moore homology theories
5.2 Other oriented theories
文摘
著作權頁：
Quillen in (29) refined Milnor's and Novikov's computations that the complex cobordism MU* of a point is a polynomial algebra with integral coefficient by showing that the map is an isomorphism (here we mean that φtop double the degrees and that the odd part of MU* varushes).Then in , Quillen produced a geometric proof of that fact emphasizing that MU* is the universal complex oriented cohomology theory on the category of differentiable manifolds.
The theorem of Conner-Floyd (5) now asserts that for each CW-complex X the map is not an isomorphism (not even surjective), even when restricted to the even part.Thus contrary to theorem 1.2.18, theorem 1.2.19 has no obvious counterpart in topology.
To give a heuristic explanation of our results we should mention that for smooth varieties over a field singular cohomology is replaced by motivic cohomology H*,*(X;Z), complex K-theory by Quillen's algebraic K-theory K *,*(X ) and complex cobordism by the theory M GL*,* represented by the algebraic Thom complex M GL (in the setting of A1-homotopy theory, see.One should note that these theories take values in the category of bigraded rings, the first degree corresponding to the cohomological degree and the second to the weight.In this setting, one should still have the Conner-Floyd isomorphism3
for any simplicial smooth k-variety X (beware here that β has bidegree (-2,-1)).However the map MGL*,*(X) L* Z→ H*,*(X) would almost never be an isomorphism.Instead one expects a spectral sequence4 from motivic cohomology to MGL*,*(X), the filtration considered in ξ4.5.2 should by the way be the one induced by that spectral sequence.Then theorem 1.2.19 is explained by the degeneration of this spectral sequence in the area computing the bidegrees of the form (2n.,n,).
Quillen in (29) refined Milnor's and Novikov's computations that the complex cobordism MU* of a point is a polynomial algebra with integral coefficient by showing that the map is an isomorphism (here we mean that φtop double the degrees and that the odd part of MU* varushes).Then in , Quillen produced a geometric proof of that fact emphasizing that MU* is the universal complex oriented cohomology theory on the category of differentiable manifolds.
The theorem of Conner-Floyd (5) now asserts that for each CW-complex X the map is not an isomorphism (not even surjective), even when restricted to the even part.Thus contrary to theorem 1.2.18, theorem 1.2.19 has no obvious counterpart in topology.
To give a heuristic explanation of our results we should mention that for smooth varieties over a field singular cohomology is replaced by motivic cohomology H*,*(X;Z), complex K-theory by Quillen's algebraic K-theory K *,*(X ) and complex cobordism by the theory M GL*,* represented by the algebraic Thom complex M GL (in the setting of A1-homotopy theory, see.One should note that these theories take values in the category of bigraded rings, the first degree corresponding to the cohomological degree and the second to the weight.In this setting, one should still have the Conner-Floyd isomorphism3
for any simplicial smooth k-variety X (beware here that β has bidegree (-2,-1)).However the map MGL*,*(X) L* Z→ H*,*(X) would almost never be an isomorphism.Instead one expects a spectral sequence4 from motivic cohomology to MGL*,*(X), the filtration considered in ξ4.5.2 should by the way be the one induced by that spectral sequence.Then theorem 1.2.19 is explained by the degeneration of this spectral sequence in the area computing the bidegrees of the form (2n.,n,).
