# Bott periodicity and exotic spheres: some comments

## 1 The Bott periodicity Theorem

One of the fundamental results in modern topology is Bott periodicity [Bott]. It computes the homotopy groups of the stable orthogonal or unitary groups $O= \cup_n O(n)$${{Stub}} ==The Bott periodicity Theorem== ; One of the fundamental results in modern topology is Bott periodicity \cite{Bott}. It computes the homotopy groups of the stable orthogonal or unitary groups O= \cup_n O(n) resp U = \cup _n U_n . The answer is very simple: For i \ge 0 one has \pi _i(O) \cong \pi_{i+8} (O) and for i>0 one has \pi_i(U) \cong \pi_{i+2}(U). Thus one only needs to know the groups for small i: For i = 1,....,8 one has \pi_{i-1}(O) \cong \mathbb Z/2, \mathbb Z/2,0,\mathbb Z,0,0,0,\mathbb Z and \pi_1(U) \cong \mathbb Z, \pi_2(U) \cong 0. Bott does not give any reference for these computations. == An interpretation in terms of vector bundles over spheres== ; Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if f : S^{i-1} \to O(n) is a continuous map, then one obtains a vector bundle E_f over S^i= D^i \cup D^i by taking two copies of D^i \times \mathbb R^n and by identifying (x,v) \in S^{i-1} \times \mathbb R^n in the first copy with (x, f(x)v) in the second copy. This map gives an isomorphism from \pi_{i-1}(O(n)) to the set Vect^\mathbb R_n(S^i) of isomorphism classes of n-dimensional vector bundles over S^i. I don't know who observed this first, it can for example be found in Steenrod's book \cite{Steenrod}. Similarly \pi_{i-1}(U(n)) corresponds to the set Vect^\mathbb C_n(S^i) of isomorphism classes of complex n-dimensional vector bundles over S^i. Passing from O(n) to O(n+1) (or U(n) to U(n+1)) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the O= \cup_n O(n)$ resp $U = \cup _n U_n$$U = \cup _n U_n$. The answer is very simple: For $i \ge 0$$i \ge 0$ one has $\pi _i(O) \cong \pi_{i+8} (O)$$\pi _i(O) \cong \pi_{i+8} (O)$ and for $i>0$$i>0$ one has $\pi_i(U) \cong \pi_{i+2}(U)$$\pi_i(U) \cong \pi_{i+2}(U)$. Thus one only needs to know the groups for small $i$$i$: For $i = 1,....,8$$i = 1,....,8$ one has
$\displaystyle \pi_{i-1}(O) \cong \mathbb Z/2, \mathbb Z/2,0,\mathbb Z,0,0,0,\mathbb Z$
and
$\displaystyle \pi_1(U) \cong \mathbb Z, \pi_2(U) \cong 0.$
Bott does not give any reference for these computations.

## 2 An interpretation in terms of vector bundles over spheres

Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if $f : S^{i-1} \to O(n)$$f : S^{i-1} \to O(n)$ is a continuous map, then one obtains a vector bundle $E_f$$E_f$ over $S^i= D^i \cup D^i$$S^i= D^i \cup D^i$ by taking two copies of $D^i \times \mathbb R^n$$D^i \times \mathbb R^n$ and by identifying $(x,v) \in S^{i-1} \times \mathbb R^n$$(x,v) \in S^{i-1} \times \mathbb R^n$ in the first copy with $(x, f(x)v)$$(x, f(x)v)$ in the second copy. This map gives an isomorphism from $\pi_{i-1}(O(n))$$\pi_{i-1}(O(n))$ to the set $Vect^\mathbb R_n(S^i)$$Vect^\mathbb R_n(S^i)$ of isomorphism classes of $n$$n$-dimensional vector bundles over $S^i$$S^i$. I don't know who observed this first, it can for example be found in Steenrod's book [Steenrod]. Similarly $\pi_{i-1}(U(n))$$\pi_{i-1}(U(n))$ corresponds to the set $Vect^\mathbb C_n(S^i)$$Vect^\mathbb C_n(S^i)$ of isomorphism classes of complex $n$$n$-dimensional vector bundles over $S^i$$S^i$. Passing from $O(n)$$O(n)$ to $O(n+1)$$O(n+1)$ (or $U(n)$$U(n)$ to $U(n+1)$$U(n+1)$) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the $1$$1$-dimensional trivial bundle. By a general position argument the stabilization map is a bijection if $n >i$$n >i$ ($n>2i$$n>2i$ in the complex case). If $n>i$$n>i$ (or $n > 2i$$n > 2i$ in the complex case) one calls such a bundle a stable vector bundle. Actually the $n$$n$-dimensional vector bundles over $S^i$$S^i$ (not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for $i>0$$i>0$ not equal to $1,2,4,8$$1,2,4,8$ mod $8$$8$ all real vector bundles of dimension $n >i$$n >i$ over $S^i$$S^i$ are trivial, that for $i=1,2$$i=1,2$ mod $8$$8$ there are precisely $2$$2$ such bundles and for $i =0$$i =0$ mod $4$$4$ there are countably many such bundles.

Remark 2.1. I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this.

## 3 The dates and dates of background papers

The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [Bott1957]. The methods of the proof were developed in several earlier papers [Bott1954] \cite {Bott 1956}.;

## 4 The role of Bott periodicity for Kervaire-Milnor's paper

Remark: {\em I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem to manifolds. In this article I would like to explain, which role the theorem plays in the paper mentioned in the title [Kervaire/Milnor]. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology. The fact that it is comparatively easy to determine the stable vector bundles over spheres (and so over any space homotopy equivalent to a sphere) suggests to ask for a smooth manifold homotopy equivalent to a sphere, a homotopy sphere, whether its stable tangent bundle (meaning one adds the trivial line bundle to the tangent bundle) is the same as for the ordinary sphere, namely trivial. If not, the homotopy spheres cannot be diffeomorphic to the standard sphere.}\\ Before Bott's theorem one probably had no chance to answer the question whether the stable tangent bundle of a homotopy sphere is trivial, with it, it is in half of the cases a triviality, since for $i = 3, 5,6,7$$i = 3, 5,6,7$ mod $8$$8$ there is no non-trivial stable vector bundle over $S^i$$S^i$. The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over $S^i$$S^i$ are isomorphic. Let's begin with the case $i = 4s$$i = 4s$. \\ Remark: {\em In the case $i=4s$$i=4s$ one has an invariant for stable vector bundles $E$$E$, namely the Pontrjagin classes $p_s(E)\in H^{4s}(S^i)$$p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the stable vector bundles over $S^{4s}$$S^{4s}$ to $H^{4s }(S^{4s})$$H^{4s }(S^{4s})$ which by choosing an orientation we identify with $\mathbb Z$$\mathbb Z$. Thus we have a homomorphism from a group isomorphic to $\mathbb Z$$\mathbb Z$ to $\mathbb Z$$\mathbb Z$, and if it is non-trivial it is an injection implying that two stable vector bundles over $S^{4s}$$S^{4s}$ are isomorphic if and only their Pontrjagin classes $p_s$$p_s$ agree. To show that the homomorphism is non-trivial (for $s>0$$s>0$) one needs a single example where this is the case. Again periodicity helps which reduces the problem to the case of bundles over $S^4$$S^4$ and $S^8$$S^8$, where one can take the tautological bundle over $S^4$$S^4$ considered as the projective line over the quaternions or over $S^8$$S^8$ considered as the projective line over the Cayley numbers. }\\ What I described in the remark is not mentioned in Kervaire-Milnor's paper. They proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class $p_s$$p_s$ are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a $4s$$4s$-dimensional homotopy sphere $\Sigma$$\Sigma$ is trivial by one sentence
"But by the Hirzebruch signature theorem the Pontrjagin class $p_k(\Sigma)$$p_k(\Sigma)$ is a multiple of the signature $\sigma (\Sigma)$$\sigma (\Sigma)$, which is zero since $H^{2s}(\Sigma) = 0$$H^{2s}(\Sigma) = 0$."\\ Remark: {\em Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated. But one sees that besides Bott's theorem another big theorem is needed to argue here.}\\ A non-trivial theorem is also needed in the remaining cases where $i = 1,2$$i = 1,2$ mod $8$$8$ is. The cases $1$$1$ and $2$$2$ are trivial, but the higher dimensions not. Here the obstruction class sits in $\pi_{i-1}(O) = \mathbb Z/2$$\pi_{i-1}(O) = \mathbb Z/2$ (by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form $0$$0$. There is a homomorphism introduced by Hopf-Whitehead, the $J$$J$-homomorphism, from $\pi_{i-1} (O)$$\pi_{i-1} (O)$ to the stable homotopy groups $\pi_{i-1}^s$$\pi_{i-1}^s$ of spheres. Rohlin has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them \cite {Kervaire-Milnor 1958}). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams \cite {Adams} saying that this $J$$J$-homomorphism is injective. Thus we summarize:

Theorem 4.1. (Theorem 3.1 [Kervaire-Milnor]) For all homotopy spheres $\Sigma$$\Sigma$ the stable tangent bundle is trivial.

;

{\em After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere. Note that - as Kervaire and Milnor mention - a homotopy sphere is, if the Poincaré conjeture is assumed, homeomorphic to the sphere and so the diffeomorphism classes of homotopy spheres would correspond to the diffemorphism classes of smooth structures on the sphere. The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for $i \ge 5$$i \ge 5$ already before the paper by Kervaire and Milnor, and later by Freedman [Freedman] in dimension $4$$4$ and by Perelmann [Perelman] in dimension $3$$3$. On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved. On the other hand the theorem allows to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension $>4$$>4$, a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called surgery theory. But the following can be said. Suppose that a topological manifold $M$$M$ is given. Then one can ask whether $M$$M$ admits a smooth structure. A necessary condition is that $M$$M$ has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension $>4$$>4$, but false in dimension $4$$4$ by the fundamental work of Donaldson, and again true in dimension $<4$$<4$ by different methods. If we assume that $M$$M$ has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension $>4$$>4$). Roughly speaking the different smooth structures on $M$$M$ correspond to the different ways to impose a stable tangent bundle on $M$$M$. Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called $K$$K$-theory, invented by Atiyah and Hirzebruch [Atiyah-Hirzebruch]. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.} \end {document};

$-dimensional trivial bundle. By a general position argument the stabilization map is a bijection if$n >i$($n>2i$in the complex case). If$n>i$(or$n > 2i$in the complex case) one calls such a bundle a '''stable vector bundle'''. Actually the$n$-dimensional vector bundles over$S^i$(not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for$i>0$not equal to O= \cup_n O(n) resp $U = \cup _n U_n$$U = \cup _n U_n$. The answer is very simple: For $i \ge 0$$i \ge 0$ one has $\pi _i(O) \cong \pi_{i+8} (O)$$\pi _i(O) \cong \pi_{i+8} (O)$ and for $i>0$$i>0$ one has $\pi_i(U) \cong \pi_{i+2}(U)$$\pi_i(U) \cong \pi_{i+2}(U)$. Thus one only needs to know the groups for small $i$$i$: For $i = 1,....,8$$i = 1,....,8$ one has $\displaystyle \pi_{i-1}(O) \cong \mathbb Z/2, \mathbb Z/2,0,\mathbb Z,0,0,0,\mathbb Z$ and $\displaystyle \pi_1(U) \cong \mathbb Z, \pi_2(U) \cong 0.$ Bott does not give any reference for these computations. ## 2 An interpretation in terms of vector bundles over spheres Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if $f : S^{i-1} \to O(n)$$f : S^{i-1} \to O(n)$ is a continuous map, then one obtains a vector bundle $E_f$$E_f$ over $S^i= D^i \cup D^i$$S^i= D^i \cup D^i$ by taking two copies of $D^i \times \mathbb R^n$$D^i \times \mathbb R^n$ and by identifying $(x,v) \in S^{i-1} \times \mathbb R^n$$(x,v) \in S^{i-1} \times \mathbb R^n$ in the first copy with $(x, f(x)v)$$(x, f(x)v)$ in the second copy. This map gives an isomorphism from $\pi_{i-1}(O(n))$$\pi_{i-1}(O(n))$ to the set $Vect^\mathbb R_n(S^i)$$Vect^\mathbb R_n(S^i)$ of isomorphism classes of $n$$n$-dimensional vector bundles over $S^i$$S^i$. I don't know who observed this first, it can for example be found in Steenrod's book [Steenrod]. Similarly $\pi_{i-1}(U(n))$$\pi_{i-1}(U(n))$ corresponds to the set $Vect^\mathbb C_n(S^i)$$Vect^\mathbb C_n(S^i)$ of isomorphism classes of complex $n$$n$-dimensional vector bundles over $S^i$$S^i$. Passing from $O(n)$$O(n)$ to $O(n+1)$$O(n+1)$ (or $U(n)$$U(n)$ to $U(n+1)$$U(n+1)$) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the $1$$1$-dimensional trivial bundle. By a general position argument the stabilization map is a bijection if $n >i$$n >i$ ($n>2i$$n>2i$ in the complex case). If $n>i$$n>i$ (or $n > 2i$$n > 2i$ in the complex case) one calls such a bundle a stable vector bundle. Actually the $n$$n$-dimensional vector bundles over $S^i$$S^i$ (not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for $i>0$$i>0$ not equal to $1,2,4,8$$1,2,4,8$ mod $8$$8$ all real vector bundles of dimension $n >i$$n >i$ over $S^i$$S^i$ are trivial, that for $i=1,2$$i=1,2$ mod $8$$8$ there are precisely $2$$2$ such bundles and for $i =0$$i =0$ mod $4$$4$ there are countably many such bundles. Remark 2.1. I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this. ## 3 The dates and dates of background papers The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [Bott1957]. The methods of the proof were developed in several earlier papers [Bott1954] \cite {Bott 1956}.; ## 4 The role of Bott periodicity for Kervaire-Milnor's paper Remark: {\em I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem to manifolds. In this article I would like to explain, which role the theorem plays in the paper mentioned in the title [Kervaire/Milnor]. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology. The fact that it is comparatively easy to determine the stable vector bundles over spheres (and so over any space homotopy equivalent to a sphere) suggests to ask for a smooth manifold homotopy equivalent to a sphere, a homotopy sphere, whether its stable tangent bundle (meaning one adds the trivial line bundle to the tangent bundle) is the same as for the ordinary sphere, namely trivial. If not, the homotopy spheres cannot be diffeomorphic to the standard sphere.}\\ Before Bott's theorem one probably had no chance to answer the question whether the stable tangent bundle of a homotopy sphere is trivial, with it, it is in half of the cases a triviality, since for $i = 3, 5,6,7$$i = 3, 5,6,7$ mod $8$$8$ there is no non-trivial stable vector bundle over $S^i$$S^i$. The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over $S^i$$S^i$ are isomorphic. Let's begin with the case $i = 4s$$i = 4s$. \\ Remark: {\em In the case $i=4s$$i=4s$ one has an invariant for stable vector bundles $E$$E$, namely the Pontrjagin classes $p_s(E)\in H^{4s}(S^i)$$p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the stable vector bundles over $S^{4s}$$S^{4s}$ to $H^{4s }(S^{4s})$$H^{4s }(S^{4s})$ which by choosing an orientation we identify with $\mathbb Z$$\mathbb Z$. Thus we have a homomorphism from a group isomorphic to $\mathbb Z$$\mathbb Z$ to $\mathbb Z$$\mathbb Z$, and if it is non-trivial it is an injection implying that two stable vector bundles over $S^{4s}$$S^{4s}$ are isomorphic if and only their Pontrjagin classes $p_s$$p_s$ agree. To show that the homomorphism is non-trivial (for $s>0$$s>0$) one needs a single example where this is the case. Again periodicity helps which reduces the problem to the case of bundles over $S^4$$S^4$ and $S^8$$S^8$, where one can take the tautological bundle over $S^4$$S^4$ considered as the projective line over the quaternions or over $S^8$$S^8$ considered as the projective line over the Cayley numbers. }\\ What I described in the remark is not mentioned in Kervaire-Milnor's paper. They proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class $p_s$$p_s$ are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a $4s$$4s$-dimensional homotopy sphere $\Sigma$$\Sigma$ is trivial by one sentence "But by the Hirzebruch signature theorem the Pontrjagin class $p_k(\Sigma)$$p_k(\Sigma)$ is a multiple of the signature $\sigma (\Sigma)$$\sigma (\Sigma)$, which is zero since $H^{2s}(\Sigma) = 0$$H^{2s}(\Sigma) = 0$."\\ Remark: {\em Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated. But one sees that besides Bott's theorem another big theorem is needed to argue here.}\\ A non-trivial theorem is also needed in the remaining cases where $i = 1,2$$i = 1,2$ mod $8$$8$ is. The cases $1$$1$ and $2$$2$ are trivial, but the higher dimensions not. Here the obstruction class sits in $\pi_{i-1}(O) = \mathbb Z/2$$\pi_{i-1}(O) = \mathbb Z/2$ (by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form $0$$0$. There is a homomorphism introduced by Hopf-Whitehead, the $J$$J$-homomorphism, from $\pi_{i-1} (O)$$\pi_{i-1} (O)$ to the stable homotopy groups $\pi_{i-1}^s$$\pi_{i-1}^s$ of spheres. Rohlin has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them \cite {Kervaire-Milnor 1958}). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams \cite {Adams} saying that this $J$$J$-homomorphism is injective. Thus we summarize: Theorem 4.1. (Theorem 3.1 [Kervaire-Milnor]) For all homotopy spheres $\Sigma$$\Sigma$ the stable tangent bundle is trivial. ; ## 5 Some comments {\em After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere. Note that - as Kervaire and Milnor mention - a homotopy sphere is, if the Poincaré conjeture is assumed, homeomorphic to the sphere and so the diffeomorphism classes of homotopy spheres would correspond to the diffemorphism classes of smooth structures on the sphere. The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for $i \ge 5$$i \ge 5$ already before the paper by Kervaire and Milnor, and later by Freedman [Freedman] in dimension $4$$4$ and by Perelmann [Perelman] in dimension $3$$3$. On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved. On the other hand the theorem allows to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension $>4$$>4$, a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called surgery theory. But the following can be said. Suppose that a topological manifold $M$$M$ is given. Then one can ask whether $M$$M$ admits a smooth structure. A necessary condition is that $M$$M$ has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension $>4$$>4$, but false in dimension $4$$4$ by the fundamental work of Donaldson, and again true in dimension $<4$$<4$ by different methods. If we assume that $M$$M$ has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension $>4$$>4$). Roughly speaking the different smooth structures on $M$$M$ correspond to the different ways to impose a stable tangent bundle on $M$$M$. Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called $K$$K$-theory, invented by Atiyah and Hirzebruch [Atiyah-Hirzebruch]. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.} \end {document}; ## 6 References ,2,4,8$ mod $all real vector bundles of dimension$n >i$over$S^i$are trivial, that for$i=1,2$mod$ there are precisely $such bundles and for$i =0$mod$ there are countably many such bundles. {{beginrem|Remark}} I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this. {{endrem}} ==The dates and dates of background papers== ; The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [[[#{{anchorencode:Bott1957}}|Bott1957]]]{{#RefAdd:Bott1957}}. The methods of the proof were developed in several earlier papers [[[#{{anchorencode:Bott1954}}|Bott1954]]]{{#RefAdd:Bott1954}} \cite {Bott 1956}. ; ==The role of Bott periodicity for Kervaire-Milnor's paper== ; '''Remark:''' {\em I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem to manifolds. In this article I would like to explain, which role the theorem plays in the paper mentioned in the title [[[#{{anchorencode:Kervaire/Milnor}}|Kervaire/Milnor]]]{{#RefAdd:Kervaire/Milnor}}. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology. The fact that it is comparatively easy to determine the stable vector bundles over spheres (and so over any space homotopy equivalent to a sphere) suggests to ask for a smooth manifold homotopy equivalent to a sphere, a '''homotopy sphere''', whether its stable tangent bundle (meaning one adds the trivial line bundle to the tangent bundle) is the same as for the ordinary sphere, namely trivial. If not, the homotopy spheres cannot be diffeomorphic to the standard sphere.}\ Before Bott's theorem one probably had no chance to answer the question whether the stable tangent bundle of a homotopy sphere is trivial, with it, it is in half of the cases a triviality, since for $i = 3, 5,6,7$ mod $there is no non-trivial stable vector bundle over$S^i$. The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over$S^i$are isomorphic. Let's begin with the case$i = 4s$. \ '''Remark:''' {\em In the case$i=4s$one has an invariant for stable vector bundles$E$, namely the Pontrjagin classes$p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the stable vector bundles over$S^{4s}$to$H^{4s }(S^{4s})$which by choosing an orientation we identify with$\mathbb Z$. Thus we have a homomorphism from a group isomorphic to$\mathbb Z$to$\mathbb Z$, and if it is non-trivial it is an injection implying that two stable vector bundles over$S^{4s}$are isomorphic if and only their Pontrjagin classes$p_s$agree. To show that the homomorphism is non-trivial (for$s>0$) one needs a single example where this is the case. Again periodicity helps which reduces the problem to the case of bundles over$S^4$and$S^8$, where one can take the tautological bundle over$S^4$considered as the projective line over the quaternions or over$S^8$considered as the projective line over the Cayley numbers. }\ What I described in the remark is not mentioned in Kervaire-Milnor's paper. They proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class$p_s$are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a s$-dimensional homotopy sphere $\Sigma$ is trivial by one sentence: "But by the Hirzebruch signature theorem the Pontrjagin class $p_k(\Sigma)$ is a multiple of the signature $\sigma (\Sigma)$, which is zero since $H^{2s}(\Sigma) = 0$."\ '''Remark:''' {\em Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated. But one sees that besides Bott's theorem another big theorem is needed to argue here.}\ A non-trivial theorem is also needed in the remaining cases where $i = 1,2$ mod $is. The cases O= \cup_n O(n) resp $U = \cup _n U_n$$U = \cup _n U_n$. The answer is very simple: For $i \ge 0$$i \ge 0$ one has $\pi _i(O) \cong \pi_{i+8} (O)$$\pi _i(O) \cong \pi_{i+8} (O)$ and for $i>0$$i>0$ one has $\pi_i(U) \cong \pi_{i+2}(U)$$\pi_i(U) \cong \pi_{i+2}(U)$. Thus one only needs to know the groups for small $i$$i$: For $i = 1,....,8$$i = 1,....,8$ one has $\displaystyle \pi_{i-1}(O) \cong \mathbb Z/2, \mathbb Z/2,0,\mathbb Z,0,0,0,\mathbb Z$ and $\displaystyle \pi_1(U) \cong \mathbb Z, \pi_2(U) \cong 0.$ Bott does not give any reference for these computations. ## 2 An interpretation in terms of vector bundles over spheres Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if $f : S^{i-1} \to O(n)$$f : S^{i-1} \to O(n)$ is a continuous map, then one obtains a vector bundle $E_f$$E_f$ over $S^i= D^i \cup D^i$$S^i= D^i \cup D^i$ by taking two copies of $D^i \times \mathbb R^n$$D^i \times \mathbb R^n$ and by identifying $(x,v) \in S^{i-1} \times \mathbb R^n$$(x,v) \in S^{i-1} \times \mathbb R^n$ in the first copy with $(x, f(x)v)$$(x, f(x)v)$ in the second copy. This map gives an isomorphism from $\pi_{i-1}(O(n))$$\pi_{i-1}(O(n))$ to the set $Vect^\mathbb R_n(S^i)$$Vect^\mathbb R_n(S^i)$ of isomorphism classes of $n$$n$-dimensional vector bundles over $S^i$$S^i$. I don't know who observed this first, it can for example be found in Steenrod's book [Steenrod]. Similarly $\pi_{i-1}(U(n))$$\pi_{i-1}(U(n))$ corresponds to the set $Vect^\mathbb C_n(S^i)$$Vect^\mathbb C_n(S^i)$ of isomorphism classes of complex $n$$n$-dimensional vector bundles over $S^i$$S^i$. Passing from $O(n)$$O(n)$ to $O(n+1)$$O(n+1)$ (or $U(n)$$U(n)$ to $U(n+1)$$U(n+1)$) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the $1$$1$-dimensional trivial bundle. By a general position argument the stabilization map is a bijection if $n >i$$n >i$ ($n>2i$$n>2i$ in the complex case). If $n>i$$n>i$ (or $n > 2i$$n > 2i$ in the complex case) one calls such a bundle a stable vector bundle. Actually the $n$$n$-dimensional vector bundles over $S^i$$S^i$ (not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for $i>0$$i>0$ not equal to $1,2,4,8$$1,2,4,8$ mod $8$$8$ all real vector bundles of dimension $n >i$$n >i$ over $S^i$$S^i$ are trivial, that for $i=1,2$$i=1,2$ mod $8$$8$ there are precisely $2$$2$ such bundles and for $i =0$$i =0$ mod $4$$4$ there are countably many such bundles. Remark 2.1. I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this. ## 3 The dates and dates of background papers The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [Bott1957]. The methods of the proof were developed in several earlier papers [Bott1954] \cite {Bott 1956}.; ## 4 The role of Bott periodicity for Kervaire-Milnor's paper Remark: {\em I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem to manifolds. In this article I would like to explain, which role the theorem plays in the paper mentioned in the title [Kervaire/Milnor]. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology. The fact that it is comparatively easy to determine the stable vector bundles over spheres (and so over any space homotopy equivalent to a sphere) suggests to ask for a smooth manifold homotopy equivalent to a sphere, a homotopy sphere, whether its stable tangent bundle (meaning one adds the trivial line bundle to the tangent bundle) is the same as for the ordinary sphere, namely trivial. If not, the homotopy spheres cannot be diffeomorphic to the standard sphere.}\\ Before Bott's theorem one probably had no chance to answer the question whether the stable tangent bundle of a homotopy sphere is trivial, with it, it is in half of the cases a triviality, since for $i = 3, 5,6,7$$i = 3, 5,6,7$ mod $8$$8$ there is no non-trivial stable vector bundle over $S^i$$S^i$. The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over $S^i$$S^i$ are isomorphic. Let's begin with the case $i = 4s$$i = 4s$. \\ Remark: {\em In the case $i=4s$$i=4s$ one has an invariant for stable vector bundles $E$$E$, namely the Pontrjagin classes $p_s(E)\in H^{4s}(S^i)$$p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the stable vector bundles over $S^{4s}$$S^{4s}$ to $H^{4s }(S^{4s})$$H^{4s }(S^{4s})$ which by choosing an orientation we identify with $\mathbb Z$$\mathbb Z$. Thus we have a homomorphism from a group isomorphic to $\mathbb Z$$\mathbb Z$ to $\mathbb Z$$\mathbb Z$, and if it is non-trivial it is an injection implying that two stable vector bundles over $S^{4s}$$S^{4s}$ are isomorphic if and only their Pontrjagin classes $p_s$$p_s$ agree. To show that the homomorphism is non-trivial (for $s>0$$s>0$) one needs a single example where this is the case. Again periodicity helps which reduces the problem to the case of bundles over $S^4$$S^4$ and $S^8$$S^8$, where one can take the tautological bundle over $S^4$$S^4$ considered as the projective line over the quaternions or over $S^8$$S^8$ considered as the projective line over the Cayley numbers. }\\ What I described in the remark is not mentioned in Kervaire-Milnor's paper. They proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class $p_s$$p_s$ are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a $4s$$4s$-dimensional homotopy sphere $\Sigma$$\Sigma$ is trivial by one sentence "But by the Hirzebruch signature theorem the Pontrjagin class $p_k(\Sigma)$$p_k(\Sigma)$ is a multiple of the signature $\sigma (\Sigma)$$\sigma (\Sigma)$, which is zero since $H^{2s}(\Sigma) = 0$$H^{2s}(\Sigma) = 0$."\\ Remark: {\em Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated. But one sees that besides Bott's theorem another big theorem is needed to argue here.}\\ A non-trivial theorem is also needed in the remaining cases where $i = 1,2$$i = 1,2$ mod $8$$8$ is. The cases $1$$1$ and $2$$2$ are trivial, but the higher dimensions not. Here the obstruction class sits in $\pi_{i-1}(O) = \mathbb Z/2$$\pi_{i-1}(O) = \mathbb Z/2$ (by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form $0$$0$. There is a homomorphism introduced by Hopf-Whitehead, the $J$$J$-homomorphism, from $\pi_{i-1} (O)$$\pi_{i-1} (O)$ to the stable homotopy groups $\pi_{i-1}^s$$\pi_{i-1}^s$ of spheres. Rohlin has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them \cite {Kervaire-Milnor 1958}). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams \cite {Adams} saying that this $J$$J$-homomorphism is injective. Thus we summarize: Theorem 4.1. (Theorem 3.1 [Kervaire-Milnor]) For all homotopy spheres $\Sigma$$\Sigma$ the stable tangent bundle is trivial. ; ## 5 Some comments {\em After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere. Note that - as Kervaire and Milnor mention - a homotopy sphere is, if the Poincaré conjeture is assumed, homeomorphic to the sphere and so the diffeomorphism classes of homotopy spheres would correspond to the diffemorphism classes of smooth structures on the sphere. The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for $i \ge 5$$i \ge 5$ already before the paper by Kervaire and Milnor, and later by Freedman [Freedman] in dimension $4$$4$ and by Perelmann [Perelman] in dimension $3$$3$. On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved. On the other hand the theorem allows to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension $>4$$>4$, a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called surgery theory. But the following can be said. Suppose that a topological manifold $M$$M$ is given. Then one can ask whether $M$$M$ admits a smooth structure. A necessary condition is that $M$$M$ has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension $>4$$>4$, but false in dimension $4$$4$ by the fundamental work of Donaldson, and again true in dimension $<4$$<4$ by different methods. If we assume that $M$$M$ has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension $>4$$>4$). Roughly speaking the different smooth structures on $M$$M$ correspond to the different ways to impose a stable tangent bundle on $M$$M$. Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called $K$$K$-theory, invented by Atiyah and Hirzebruch [Atiyah-Hirzebruch]. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.} \end {document}; ## 6 References$ and $are trivial, but the higher dimensions not. Here the obstruction class sits in$\pi_{i-1}(O) = \mathbb Z/2$(by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form $O= \cup_n O(n)$ resp $U = \cup _n U_n$$U = \cup _n U_n$. The answer is very simple: For $i \ge 0$$i \ge 0$ one has $\pi _i(O) \cong \pi_{i+8} (O)$$\pi _i(O) \cong \pi_{i+8} (O)$ and for $i>0$$i>0$ one has $\pi_i(U) \cong \pi_{i+2}(U)$$\pi_i(U) \cong \pi_{i+2}(U)$. Thus one only needs to know the groups for small $i$$i$: For $i = 1,....,8$$i = 1,....,8$ one has $\displaystyle \pi_{i-1}(O) \cong \mathbb Z/2, \mathbb Z/2,0,\mathbb Z,0,0,0,\mathbb Z$ and $\displaystyle \pi_1(U) \cong \mathbb Z, \pi_2(U) \cong 0.$ Bott does not give any reference for these computations. ## 2 An interpretation in terms of vector bundles over spheres Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if $f : S^{i-1} \to O(n)$$f : S^{i-1} \to O(n)$ is a continuous map, then one obtains a vector bundle $E_f$$E_f$ over $S^i= D^i \cup D^i$$S^i= D^i \cup D^i$ by taking two copies of $D^i \times \mathbb R^n$$D^i \times \mathbb R^n$ and by identifying $(x,v) \in S^{i-1} \times \mathbb R^n$$(x,v) \in S^{i-1} \times \mathbb R^n$ in the first copy with $(x, f(x)v)$$(x, f(x)v)$ in the second copy. This map gives an isomorphism from $\pi_{i-1}(O(n))$$\pi_{i-1}(O(n))$ to the set $Vect^\mathbb R_n(S^i)$$Vect^\mathbb R_n(S^i)$ of isomorphism classes of $n$$n$-dimensional vector bundles over $S^i$$S^i$. I don't know who observed this first, it can for example be found in Steenrod's book [Steenrod]. Similarly $\pi_{i-1}(U(n))$$\pi_{i-1}(U(n))$ corresponds to the set $Vect^\mathbb C_n(S^i)$$Vect^\mathbb C_n(S^i)$ of isomorphism classes of complex $n$$n$-dimensional vector bundles over $S^i$$S^i$. Passing from $O(n)$$O(n)$ to $O(n+1)$$O(n+1)$ (or $U(n)$$U(n)$ to $U(n+1)$$U(n+1)$) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the $1$$1$-dimensional trivial bundle. By a general position argument the stabilization map is a bijection if $n >i$$n >i$ ($n>2i$$n>2i$ in the complex case). If $n>i$$n>i$ (or $n > 2i$$n > 2i$ in the complex case) one calls such a bundle a stable vector bundle. Actually the $n$$n$-dimensional vector bundles over $S^i$$S^i$ (not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for $i>0$$i>0$ not equal to $1,2,4,8$$1,2,4,8$ mod $8$$8$ all real vector bundles of dimension $n >i$$n >i$ over $S^i$$S^i$ are trivial, that for $i=1,2$$i=1,2$ mod $8$$8$ there are precisely $2$$2$ such bundles and for $i =0$$i =0$ mod $4$$4$ there are countably many such bundles. Remark 2.1. I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this. ## 3 The dates and dates of background papers The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [Bott1957]. The methods of the proof were developed in several earlier papers [Bott1954] \cite {Bott 1956}.; ## 4 The role of Bott periodicity for Kervaire-Milnor's paper Remark: {\em I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem to manifolds. In this article I would like to explain, which role the theorem plays in the paper mentioned in the title [Kervaire/Milnor]. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology. The fact that it is comparatively easy to determine the stable vector bundles over spheres (and so over any space homotopy equivalent to a sphere) suggests to ask for a smooth manifold homotopy equivalent to a sphere, a homotopy sphere, whether its stable tangent bundle (meaning one adds the trivial line bundle to the tangent bundle) is the same as for the ordinary sphere, namely trivial. If not, the homotopy spheres cannot be diffeomorphic to the standard sphere.}\\ Before Bott's theorem one probably had no chance to answer the question whether the stable tangent bundle of a homotopy sphere is trivial, with it, it is in half of the cases a triviality, since for $i = 3, 5,6,7$$i = 3, 5,6,7$ mod $8$$8$ there is no non-trivial stable vector bundle over $S^i$$S^i$. The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over $S^i$$S^i$ are isomorphic. Let's begin with the case $i = 4s$$i = 4s$. \\ Remark: {\em In the case $i=4s$$i=4s$ one has an invariant for stable vector bundles $E$$E$, namely the Pontrjagin classes $p_s(E)\in H^{4s}(S^i)$$p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the stable vector bundles over $S^{4s}$$S^{4s}$ to $H^{4s }(S^{4s})$$H^{4s }(S^{4s})$ which by choosing an orientation we identify with $\mathbb Z$$\mathbb Z$. Thus we have a homomorphism from a group isomorphic to $\mathbb Z$$\mathbb Z$ to $\mathbb Z$$\mathbb Z$, and if it is non-trivial it is an injection implying that two stable vector bundles over $S^{4s}$$S^{4s}$ are isomorphic if and only their Pontrjagin classes $p_s$$p_s$ agree. To show that the homomorphism is non-trivial (for $s>0$$s>0$) one needs a single example where this is the case. Again periodicity helps which reduces the problem to the case of bundles over $S^4$$S^4$ and $S^8$$S^8$, where one can take the tautological bundle over $S^4$$S^4$ considered as the projective line over the quaternions or over $S^8$$S^8$ considered as the projective line over the Cayley numbers. }\\ What I described in the remark is not mentioned in Kervaire-Milnor's paper. They proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class $p_s$$p_s$ are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a $4s$$4s$-dimensional homotopy sphere $\Sigma$$\Sigma$ is trivial by one sentence "But by the Hirzebruch signature theorem the Pontrjagin class $p_k(\Sigma)$$p_k(\Sigma)$ is a multiple of the signature $\sigma (\Sigma)$$\sigma (\Sigma)$, which is zero since $H^{2s}(\Sigma) = 0$$H^{2s}(\Sigma) = 0$."\\ Remark: {\em Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated. But one sees that besides Bott's theorem another big theorem is needed to argue here.}\\ A non-trivial theorem is also needed in the remaining cases where $i = 1,2$$i = 1,2$ mod $8$$8$ is. The cases $1$$1$ and $2$$2$ are trivial, but the higher dimensions not. Here the obstruction class sits in $\pi_{i-1}(O) = \mathbb Z/2$$\pi_{i-1}(O) = \mathbb Z/2$ (by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form $0$$0$. There is a homomorphism introduced by Hopf-Whitehead, the $J$$J$-homomorphism, from $\pi_{i-1} (O)$$\pi_{i-1} (O)$ to the stable homotopy groups $\pi_{i-1}^s$$\pi_{i-1}^s$ of spheres. Rohlin has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them \cite {Kervaire-Milnor 1958}). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams \cite {Adams} saying that this $J$$J$-homomorphism is injective. Thus we summarize: Theorem 4.1. (Theorem 3.1 [Kervaire-Milnor]) For all homotopy spheres $\Sigma$$\Sigma$ the stable tangent bundle is trivial. ; ## 5 Some comments {\em After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere. Note that - as Kervaire and Milnor mention - a homotopy sphere is, if the Poincaré conjeture is assumed, homeomorphic to the sphere and so the diffeomorphism classes of homotopy spheres would correspond to the diffemorphism classes of smooth structures on the sphere. The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for $i \ge 5$$i \ge 5$ already before the paper by Kervaire and Milnor, and later by Freedman [Freedman] in dimension $4$$4$ and by Perelmann [Perelman] in dimension $3$$3$. On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved. On the other hand the theorem allows to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension $>4$$>4$, a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called surgery theory. But the following can be said. Suppose that a topological manifold $M$$M$ is given. Then one can ask whether $M$$M$ admits a smooth structure. A necessary condition is that $M$$M$ has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension $>4$$>4$, but false in dimension $4$$4$ by the fundamental work of Donaldson, and again true in dimension $<4$$<4$ by different methods. If we assume that $M$$M$ has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension $>4$$>4$). Roughly speaking the different smooth structures on $M$$M$ correspond to the different ways to impose a stable tangent bundle on $M$$M$. Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called $K$$K$-theory, invented by Atiyah and Hirzebruch [Atiyah-Hirzebruch]. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.} \end {document}; ## 6 References$. There is a homomorphism introduced by Hopf-Whitehead, the $J$-homomorphism, from $\pi_{i-1} (O)$ to the stable homotopy groups $\pi_{i-1}^s$ of spheres. Rohlin has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them \cite {Kervaire-Milnor 1958}). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams \cite {Adams} saying that this $J$-homomorphism is injective. Thus we summarize: {{beginthm|Theorem|}} (Theorem 3.1 [[[#{{anchorencode:Kervaire-Milnor}}|Kervaire-Milnor]]]{{#RefAdd:Kervaire-Milnor}}) For all homotopy spheres $\Sigma$ the stable tangent bundle is trivial. {{endthm}}
; ==Some comments== ; {\em After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere. Note that - as Kervaire and Milnor mention - a homotopy sphere is, if the Poincaré conjeture is assumed, homeomorphic to the sphere and so the diffeomorphism classes of homotopy spheres would correspond to the diffemorphism classes of smooth structures on the sphere. The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for $i \ge 5$ already before the paper by Kervaire and Milnor, and later by Freedman [[[#{{anchorencode:Freedman}}|Freedman]]]{{#RefAdd:Freedman}} in dimension $and by Perelmann [[[#{{anchorencode:Perelman}}|Perelman]]]{{#RefAdd:Perelman}} in dimension$. On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved. On the other hand the theorem allows to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension $>4$, a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called '''surgery theory'''. But the following can be said. Suppose that a topological manifold $M$ is given. Then one can ask whether $M$ admits a smooth structure. A necessary condition is that $M$ has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension $>4$, but false in dimension $by the fundamental work of Donaldson, and again true in dimension$<4$by different methods. If we assume that$M$has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension$>4$). Roughly speaking the different smooth structures on$M$correspond to the different ways to impose a stable tangent bundle on$M$. Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called$K\$-theory, invented by Atiyah and Hirzebruch [[[#{{anchorencode:Atiyah-Hirzebruch}}|Atiyah-Hirzebruch]]]{{#RefAdd:Atiyah-Hirzebruch}}. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.} \end {document}; == References == {{#RefList:}} [[Category:History]]O= \cup_n O(n) resp $U = \cup _n U_n$$U = \cup _n U_n$. The answer is very simple: For $i \ge 0$$i \ge 0$ one has $\pi _i(O) \cong \pi_{i+8} (O)$$\pi _i(O) \cong \pi_{i+8} (O)$ and for $i>0$$i>0$ one has $\pi_i(U) \cong \pi_{i+2}(U)$$\pi_i(U) \cong \pi_{i+2}(U)$. Thus one only needs to know the groups for small $i$$i$: For $i = 1,....,8$$i = 1,....,8$ one has
$\displaystyle \pi_{i-1}(O) \cong \mathbb Z/2, \mathbb Z/2,0,\mathbb Z,0,0,0,\mathbb Z$
and
$\displaystyle \pi_1(U) \cong \mathbb Z, \pi_2(U) \cong 0.$
Bott does not give any reference for these computations.

## 2 An interpretation in terms of vector bundles over spheres

Most applications of this result concern the interpretation of these groups as stable vector bundles over spheres. Namely if $f : S^{i-1} \to O(n)$$f : S^{i-1} \to O(n)$ is a continuous map, then one obtains a vector bundle $E_f$$E_f$ over $S^i= D^i \cup D^i$$S^i= D^i \cup D^i$ by taking two copies of $D^i \times \mathbb R^n$$D^i \times \mathbb R^n$ and by identifying $(x,v) \in S^{i-1} \times \mathbb R^n$$(x,v) \in S^{i-1} \times \mathbb R^n$ in the first copy with $(x, f(x)v)$$(x, f(x)v)$ in the second copy. This map gives an isomorphism from $\pi_{i-1}(O(n))$$\pi_{i-1}(O(n))$ to the set $Vect^\mathbb R_n(S^i)$$Vect^\mathbb R_n(S^i)$ of isomorphism classes of $n$$n$-dimensional vector bundles over $S^i$$S^i$. I don't know who observed this first, it can for example be found in Steenrod's book [Steenrod]. Similarly $\pi_{i-1}(U(n))$$\pi_{i-1}(U(n))$ corresponds to the set $Vect^\mathbb C_n(S^i)$$Vect^\mathbb C_n(S^i)$ of isomorphism classes of complex $n$$n$-dimensional vector bundles over $S^i$$S^i$. Passing from $O(n)$$O(n)$ to $O(n+1)$$O(n+1)$ (or $U(n)$$U(n)$ to $U(n+1)$$U(n+1)$) by the standard inclusion corresponds to stabilization of vector bundles by taking the Whitney sum with the $1$$1$-dimensional trivial bundle. By a general position argument the stabilization map is a bijection if $n >i$$n >i$ ($n>2i$$n>2i$ in the complex case). If $n>i$$n>i$ (or $n > 2i$$n > 2i$ in the complex case) one calls such a bundle a stable vector bundle. Actually the $n$$n$-dimensional vector bundles over $S^i$$S^i$ (not over a general space) form an abelian group, where the sum is given by a connected sum of vector bundles: One choses a trivialization of the vector bundles over an open disk and identifies the resulting boundaries. Bott's theorem implies that for $i>0$$i>0$ not equal to $1,2,4,8$$1,2,4,8$ mod $8$$8$ all real vector bundles of dimension $n >i$$n >i$ over $S^i$$S^i$ are trivial, that for $i=1,2$$i=1,2$ mod $8$$8$ there are precisely $2$$2$ such bundles and for $i =0$$i =0$ mod $4$$4$ there are countably many such bundles.

Remark 2.1. I find it remarkable that Bott doesn't mention the relation to vector bundles. Actually Bott does not say a single word, why his result is interesting. He obviously assumes that a reader finds the problem to determine the homotopy groups of such fundamental objects like the stable orthogonal or unitary group interesting in itself and he is of course right. Whether he has foreseen that it is such a fundamental result would be interesting to know, perhaps his friends Atiyah and Hirzebruch can comment on this.

## 3 The dates and dates of background papers

The paper with complete proofs appeared in September 1959, it was submitted November 1958. An announcement containing the above statements appeared in 1957 [Bott1957]. The methods of the proof were developed in several earlier papers [Bott1954] \cite {Bott 1956}.;

## 4 The role of Bott periodicity for Kervaire-Milnor's paper

Remark: {\em I hope there will be many articles in the atlas explaining more or less immediate applications of the periodicity theorem to manifolds. In this article I would like to explain, which role the theorem plays in the paper mentioned in the title [Kervaire/Milnor]. This paper appeared May 1963, and was submitted April 1962. By that time the periodicity theorem must have been a standard tool in topology. The fact that it is comparatively easy to determine the stable vector bundles over spheres (and so over any space homotopy equivalent to a sphere) suggests to ask for a smooth manifold homotopy equivalent to a sphere, a homotopy sphere, whether its stable tangent bundle (meaning one adds the trivial line bundle to the tangent bundle) is the same as for the ordinary sphere, namely trivial. If not, the homotopy spheres cannot be diffeomorphic to the standard sphere.}\\ Before Bott's theorem one probably had no chance to answer the question whether the stable tangent bundle of a homotopy sphere is trivial, with it, it is in half of the cases a triviality, since for $i = 3, 5,6,7$$i = 3, 5,6,7$ mod $8$$8$ there is no non-trivial stable vector bundle over $S^i$$S^i$. The remaining cases are not so easy, one needs a way to decide whether two stable vector bundles over $S^i$$S^i$ are isomorphic. Let's begin with the case $i = 4s$$i = 4s$. \\ Remark: {\em In the case $i=4s$$i=4s$ one has an invariant for stable vector bundles $E$$E$, namely the Pontrjagin classes $p_s(E)\in H^{4s}(S^i)$$p_s(E)\in H^{4s}(S^i)$. It turns out that this map is a homomorphism from the stable vector bundles over $S^{4s}$$S^{4s}$ to $H^{4s }(S^{4s})$$H^{4s }(S^{4s})$ which by choosing an orientation we identify with $\mathbb Z$$\mathbb Z$. Thus we have a homomorphism from a group isomorphic to $\mathbb Z$$\mathbb Z$ to $\mathbb Z$$\mathbb Z$, and if it is non-trivial it is an injection implying that two stable vector bundles over $S^{4s}$$S^{4s}$ are isomorphic if and only their Pontrjagin classes $p_s$$p_s$ agree. To show that the homomorphism is non-trivial (for $s>0$$s>0$) one needs a single example where this is the case. Again periodicity helps which reduces the problem to the case of bundles over $S^4$$S^4$ and $S^8$$S^8$, where one can take the tautological bundle over $S^4$$S^4$ considered as the projective line over the quaternions or over $S^8$$S^8$ considered as the projective line over the Cayley numbers. }\\ What I described in the remark is not mentioned in Kervaire-Milnor's paper. They proceed slightly differently. They refer to obstruction theory, a theory which can be used to decide whether a vector bundle is trivial. They say (with reference to earlier papers by Kervaire) that the obstruction class and the Pontrjagin class $p_s$$p_s$ are proportional by a non-zero factor and so, if the Pontrjagin class is trivial, the bundle is trivial. They finish the argument that the stable tangent bundle of a $4s$$4s$-dimensional homotopy sphere $\Sigma$$\Sigma$ is trivial by one sentence
"But by the Hirzebruch signature theorem the Pontrjagin class $p_k(\Sigma)$$p_k(\Sigma)$ is a multiple of the signature $\sigma (\Sigma)$$\sigma (\Sigma)$, which is zero since $H^{2s}(\Sigma) = 0$$H^{2s}(\Sigma) = 0$."\\ Remark: {\em Let me make a short comment on this sentence. It shows that the signature theorem, which was published by Hirzebruch in his 1956 book, is at the time when Kervaire and Milnor wrote their paper so standard, that neither a reference to Hirzebruch's book is needed (which appears in the list of references but no reference is given at this place) nor the formula of the signature theorem is repeated. But one sees that besides Bott's theorem another big theorem is needed to argue here.}\\ A non-trivial theorem is also needed in the remaining cases where $i = 1,2$$i = 1,2$ mod $8$$8$ is. The cases $1$$1$ and $2$$2$ are trivial, but the higher dimensions not. Here the obstruction class sits in $\pi_{i-1}(O) = \mathbb Z/2$$\pi_{i-1}(O) = \mathbb Z/2$ (by Bott's theorem) and so again one has to find a way to distinguish the non-trivial element form $0$$0$. There is a homomorphism introduced by Hopf-Whitehead, the $J$$J$-homomorphism, from $\pi_{i-1} (O)$$\pi_{i-1} (O)$ to the stable homotopy groups $\pi_{i-1}^s$$\pi_{i-1}^s$ of spheres. Rohlin has shown that under this homomorphism the obstruction class vanishes (the authors don't give a reference to a paper by Rohlin but refer instead to an earlier paper by them \cite {Kervaire-Milnor 1958}). The argument, that also in the remaining case the stable tangent bundle is trivial is finished by applying a recent deep theorem by Adams \cite {Adams} saying that this $J$$J$-homomorphism is injective. Thus we summarize:

Theorem 4.1. (Theorem 3.1 [Kervaire-Milnor]) For all homotopy spheres $\Sigma$$\Sigma$ the stable tangent bundle is trivial.

;

{\em After repeating the role of Bott periodicity in the proof of this theorem I would like to comment a bit on the role of this theorem in the paper of Kervaire and Milnor and in the further development of analyzing smooth structures on a topological manifold, here the sphere. Note that - as Kervaire and Milnor mention - a homotopy sphere is, if the Poincaré conjeture is assumed, homeomorphic to the sphere and so the diffeomorphism classes of homotopy spheres would correspond to the diffemorphism classes of smooth structures on the sphere. The Poincaré conjecture was proven by Smale, Stallings and Zeeman (references) for $i \ge 5$$i \ge 5$ already before the paper by Kervaire and Milnor, and later by Freedman [Freedman] in dimension $4$$4$ and by Perelmann [Perelman] in dimension $3$$3$. On the one hand the message of the theorem is negative, one cannot use the stable tangent bundle to distinguish different smooth structures on the sphere. This leads to the interesting question, whether the same is true for arbitrary manifolds. Later in the sixties some very deep theorems in this direction were proved. On the other hand the theorem allows to develop a method to study the different smooth structures on spheres or equivalently the homotopy spheres in dimension $>4$$>4$, a method which in the following years was generalized to arbitrary manifolds. This is not the place to describe this method, which is called surgery theory. But the following can be said. Suppose that a topological manifold $M$$M$ is given. Then one can ask whether $M$$M$ admits a smooth structure. A necessary condition is that $M$$M$ has a tangent bundle, and since this is easier to analyze and essentially the same, that it has a stable tangent bundle. It turns out that in a certain sense, which should be made precise elsewhere, this is the only obstruction, again in dimension $>4$$>4$, but false in dimension $4$$4$ by the fundamental work of Donaldson, and again true in dimension $<4$$<4$ by different methods. If we assume that $M$$M$ has a smooth structure, we choose one and compare all possible other smooth structures with this. Then again the stable tangent bundle plays the deciding role (in dimension $>4$$>4$). Roughly speaking the different smooth structures on $M$$M$ correspond to the different ways to impose a stable tangent bundle on $M$$M$. Thus the understanding of stable vector bundles is what at the end is needed. This is the content of a very important theory, called $K$$K$-theory, invented by Atiyah and Hirzebruch [Atiyah-Hirzebruch]. This is a generalized cohomology theory (meaning that the Eilenberg-Steenrod axioms for ordinary cohomology are fulfilled except the dimension axiom). This is the first generalized cohomology theory and - besides stable homotopy - the most important one. To construct it is up to a certain point rather elementary. But then one comes at a point where the arguments are highly non-trivial and the central tool, which one has to apply, is Bott perodicity. It does not only give the fundamental input for completing the proof that it is a generalized homology theory, it also is the central tool for all computations. In particular, in the few cases where one can give detailed information about the different smooth structures on a manifold, always Bott's theorem is in the background - like we indicated in one aspect for the spheres.} \end {document};