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− | To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:
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− | - For statements like Theorem, Lemma, Definition etc., use e.g.
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− | {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.
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− | - For references, use e.g. {{cite|Milnor1958b}}.
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− | END OF COMMENT
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− | == Question ==
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| <wikitex>; | | <wikitex>; |
− | Take the stable vector bundle $\xi$ over the $4i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$. By defintion the the primary obstruction to trivialising $\xi^{4i}$ is an obstruction class $x \in H^{4i}(S^{4i})$ which generates $H^{4i}(S^{4i}) \cong \mathbb{Z}$. | + | Take the stable vector bundle $\xi$ over the $4i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$. By defintion, the the primary obstruction to trivialising $\xi^{4i}$ is a cohomology class $x \in H^{4i}(S^{4i}; \pi_{4k-1}(O)) = H^{4i}(S^{4i})$ which generates $H^{4i}(S^{4i}) \cong \mathbb{Z}$. |
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− | {{beginthm|Question}} | + | {{beginthm|Exercise}} |
− | What is the $i$-th integral Pontryagin class of $\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i})$ ?
| + | Determine the $i$-th integral Pontryagin class of $\xi^{4i}$, $p_i(\xi) \in H^{4i}(S^{4i})$, in terms of $x$. |
| + | {{endthm}} |
| </wikitex> | | </wikitex> |
− | == Answer ==
| + | [[Category:Exercises]] |
− | <wikitex>;
| + | [[Category:Exercises with solution]] |
− | Let $a_j = (3 - (-1)^j)/2$, let $k!$ be the integer k-factorial and recall that $x \in H^{4i}(S^{4i})$ is a generator.
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− | {{beginthm|Theorem|{{cite|Kervaire1959}}}} \label{thm:1}
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− | There is an identity
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− | $$ p_i(\xi^{4i}) = \pm a_i \cdot (2i-1)! \cdot x \in H^{4i}(S^{4i}).$$
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− | Similarly, if $\eta$ denotes the complex vector bundle over $S^{2i}$ corresponding to a generator of $\pi_{2i}(BU)\cong \mathbb{Z}$, then its Chern class is given by
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− | $$ c_i(\eta) = \pm (i-1)! \cdot y\in H^{2i}(S^{2i}),$$
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− | where $y\in H^{2i}(S^{2i})\cong \mathbb{Z}$ is a generator.
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− | {{endthm|Theorem}}
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− | === Justification ===
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− | A way to prove the Theorem \ref{thm:1} is to use the Chern character
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− | $$\tilde K_0(X)\to \tilde H^{ev}(X;\mathbb{Q})$$
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− | from complex topological $K$-theory. It can be defined using the explicit formula
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− | $$ ch(\xi)= \sum_{k>0} s_k(c_1(\xi),\dots,c_k(\xi))/k! $$
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− | for a virtual complex vector bundle $\xi$, where $s_k$ are the Newton polynomials. In the case $X=S^{2n}$ two special things occur:
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− | #The Chern character is injective with image $H^{2n}(S^{2n};\mathbb{Z})$. (This follows inductively from the case $n=0$ using Bott periodicity.)
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− | #A calculation shows that the image of a (virtual) complex vector bundle $\xi$ over $S^{2n}$ is given by $\pm c_n(\xi)/(n-1)!$.
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− | Hence, $c_i(\eta)$ is given by $\pm (n-1)!$ times a generator. This establishes the second part of the Theorem.
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− | The first part follows using the definition $p_i(\xi)=c_{2i}(\xi\otimes_\mathbb{R} \mathbb{C})$ together with the fact that complexification induces a map
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− | $$ - \otimes_\mathbb{R} \mathbb{C}\colon \widetilde{KO}^0(S^{4i})\to \tilde K^0(S^{4i}) $$
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− | which is given by multiplication by $a_i$, i.e. is a isomorphism in degrees $8i$ and multiplication by 2 in degrees $8i+4$.
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− | </wikitex>
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− | == Further discussion ==
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− | <wikitex>;
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− | ...
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− | </wikitex>
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− | == References ==
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− | {{#RefList:}}
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− | <!-- Please add further headings according to your needs. -->
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− | [[Category:Questions]] | + | |
− | [[Category:Study questions]] | + | |