Talk:Obstruction classes and Pontrjagin classes (Ex)

1 Answer

Let $a_j = (3 - (-1)^j)/2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}a_j = (3 - (-1)^j)/2$, let $k!$$k!$ be the integer k-factorial and recall that $x \in H^{4i}(S^{4i})$$x \in H^{4i}(S^{4i})$ is a generator.

Theorem 1.1 [Kervaire1959]. There is an identity

$\displaystyle p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$ $$\displaystyle p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$

Similarly, if $\eta$$\eta$ denotes the complex vector bundle over $S^{2i}$$S^{2i}$ corresponding to a generator of $\pi_{2i}(BU)\cong \mathbb{Z}$$\pi_{2i}(BU)\cong \mathbb{Z}$, then its Chern class is given by

$\displaystyle c_i(\eta) = \pm (i-1)! \cdot y\in H^{2i}(S^{2i}),$ $$\displaystyle c_i(\eta) = \pm (i-1)! \cdot y\in H^{2i}(S^{2i}),$$

where $y\in H^{2i}(S^{2i})\cong \mathbb{Z}$$y\in H^{2i}(S^{2i})\cong \mathbb{Z}$ is a generator.

Justification

A way to prove the Theorem 1.1 is to use the Chern character

$\displaystyle \tilde K_0(X)\to \tilde H^{ev}(X;\mathbb{Q})$ $$\displaystyle \tilde K_0(X)\to \tilde H^{ev}(X;\mathbb{Q})$$

from complex topological $K$$K$-theory. It can be defined using the explicit formula

$\displaystyle ch(\xi)= \sum_{k>0} s_k(c_1(\xi),\dots,c_k(\xi))/k!$ $$\displaystyle ch(\xi)= \sum_{k>0} s_k(c_1(\xi),\dots,c_k(\xi))/k!$$

for a virtual complex vector bundle $\xi$$\xi$, where $s_k$$s_k$ are the Newton polynomials. In the case $X=S^{2n}$$X=S^{2n}$ two special things occur:

1. The Chern character is injective with image $H^{2n}(S^{2n};\mathbb{Q})$$H^{2n}(S^{2n};\mathbb{Q})$. This follows from the case $n=1$$n=1$ using Bott periodicity and multiplicativity:
   $\displaystyle \tilde K_0(S^{2n}) \cong\tilde K_0(S^2)\otimes \dots \otimes\tilde K_0(S^2) \to \tilde H^{ev}(S^{2};\mathbb{Q})\otimes \dots \otimes \tilde H^{ev}(S^{2};\mathbb{Q})\cong \tilde H^{ev}(S^{2n};\mathbb{Q})$ $$\displaystyle \tilde K_0(S^{2n}) \cong\tilde K_0(S^2)\otimes \dots \otimes\tilde K_0(S^2) \to \tilde H^{ev}(S^{2};\mathbb{Q})\otimes \dots \otimes \tilde H^{ev}(S^{2};\mathbb{Q})\cong \tilde H^{ev}(S^{2n};\mathbb{Q})$$
2. A calculation shows that the image of a (virtual) complex vector bundle $\xi$$\xi$ over $S^{2n}$$S^{2n}$ is given by:

$\displaystyle ch(\xi)= s_n(0,\dots, 0, c_n(\xi))/n! = \pm c_n(\xi)/(n-1)!$ $$\displaystyle ch(\xi)= s_n(0,\dots, 0, c_n(\xi))/n! = \pm c_n(\xi)/(n-1)!$$

Hence, $c_i(\eta)$$c_i(\eta)$ is given by $\pm (n-1)!$$\pm (n-1)!$ times a generator. This establishes the second part of the Theorem.

The first part follows using the definition $p_i(\xi)= (-1)^ic_{2i}(\xi\otimes_\mathbb{R} \mathbb{C})$$p_i(\xi)= (-1)^ic_{2i}(\xi\otimes_\mathbb{R} \mathbb{C})$ together with the fact that complexification induces a map

$\displaystyle - \otimes_\mathbb{R} \mathbb{C}\colon \widetilde{KO}^0(S^{4i})\to \tilde K^0(S^{4i})$ $$\displaystyle - \otimes_\mathbb{R} \mathbb{C}\colon \widetilde{KO}^0(S^{4i})\to \tilde K^0(S^{4i})$$

which is given by multiplication by $a_i$$a_i$, i.e. is a isomorphism in degrees $8i$$8i$ and multiplication by 2 in degrees $8i+4$$8i+4$.

2 Further discussion

The integrality condition for the Chern character (and the additional factor of 2 for complexifications of real vector bundles in dimensions $8i+4$$8i+4$) also follows from the Atiyah-Singer Index Theorem.

3 Desiderata for this page

Ideally this page will also include at least the following information:

1. A reference, probably to Baues' book, for obstructions to null-homotopy.
2. A precise reference to [Kervaire1959].
3. A precise reference to Bott who also proved Theorem 2.1 (perhaps even before Kervaire?)
4. Conventions/constructions for settling the sign in Theorem 2.1.
5. A reference to Husemoller's Fibre Bundles for a similar result for Chern classes. This also gives an explanation of the result via the Chern character which should be included in the page.
6. Further discussion about the same problem for other characteristic classes and other bases spaces.

Diarmuid Crowley 16:21, 17 March 2010 (UTC)

4 References

• [Kervaire1959] M. A. Kervaire, A note on obstructions and characteristic classes, Amer. J. Math. 81 (1959), 773–784. MR0107863 (21 #6585) Zbl 0124.16302