 more session of the math associate seminar of the ICTP. So our speaker today is José Luis Isneros from Dunham and he's talking about exotic spheres and singularities so please go ahead. Okay well thank you very much to Alejandra and Saq for inviting me to give this talk in the in this seminar. It's nice to be back in ICTP at this virtually and I will talk about exotic spheres and singularities and the talk is as in the abstract said about the Milner Fibration theorem which is an important theorem in singularity theory but the origin of this theorem has a curious history because it didn't start in singularity theory it started in the in differential topology no in the in the search of exotic spheres so I will tell how this theorem came to life in search of these exotic spheres so this talk is not at all technical it's just presenting the main results which lead to this important theorem. Okay so I start talking about exotic spheres and I start with the spheres the standard and sphere as we all know is just the set of points in r and plus one we satisfy this equation which geometrically means just all the points with whose distance to the origin is one. Well we have here the standard one sphere in the circle the standard two sphere and well the standard sphere is anthropological space and also a smooth manifold this will be important and anthropological end sphere is anthropological space which is homeomorphic to the standard sphere so for instance we have here the Koch snowflake which was one of the first fractals known and it appears in this paper on a continuous curve without tangents constructible from elementary geometry by Koch in 1904 and well so it doesn't have at any point any tangent so that implies that it's not diffeomorphic to the circle but it's homeomorphic so we have these two categories the topological category and the smooth category and well it's homomorphic but not diffeomorphic and well two more definitions a homotopian sphere is an end manifold which is homotopically equivalent to the standard sphere and a homology end sphere is also an end manifold which has the homology of the standard sphere that means and dimension zero is set all the other homology groups are zero except the top one not a dimension n which is also the integers and in 1900 Poincare claimed that the homology characterized the three manifolds which are homeomorphic to the three-dimensional standard sphere so in other words that every homology three sphere is homeomorphic to S3 but in 1904 he constructed a contract sample which is a three manifold of the form the three-dimensional sphere modulo the action of some finite subgroup which is now known as the Poincare homology sphere which fundamental group is this finite subgroup of S3 which is the binary hycosahedral group which is some sort of double cover of the hycosahedral group the the group of rotations of a hot hycosahedron and then Poincare asked no let it let me be a close three manifold is it possible that the fundamental group is trivial even though B is not homeomorphic to S3 in fact to when Poincare found his contract sample it was one of the motivations to invent the fundamental group and well this Poincare conjecture this question can be asked in any dimensions not not only in dimension three and now we have an answer to this question which is topological characterization of the spheres which is the generalized Poincare conjecture which says the following that the following three statements are equivalent a close n manifold m n is homeomorphic to the sphere is equivalent to that m is a homotopy n sphere or that m has the same homology and fundamental group as the standard sphere so for dimensions one and two this is a classical result then it was proved by dimensions five or higher by Smale, Stallings and Simon and dimension four by Friedmann and last by the n dimension three by Perelman which was the original Poincare conjecture so in the topological category we have a complete answer of which manifolds are homeomorphic to the standard sphere well if it satisfies any of these two conditions not to be homotopy equivalent to the n sphere or to have the same homology and fundamental group now we can concentrate in the smooth category you know we can ask which smooth manifolds are diffeomorphic to the standard sphere so for instance we have smooth knots they are diffeomorphic to the circle and of course if we have a manifold which is diffeomorphic to the sphere they are homeomorphic and therefore for instance have the same homotopy type of the sphere and one can ask the converse no if we have a manifold which is homeomorphic to the standard sphere are they diffeomorphic can we give a differentiable structure to m such that they are diffeomorphic and the question is reasonable because we have the following lemma that any homomorphism from m to sn can be uniformly approximated by the smooth maps this is basically bias trust approximation theorem and one can ask no can home homomorphism from m to sn always be approximated by diffeomorphisms so in other words can we move or modify a bit the the the proof of this lemma in order to get diffeomorphisms well the in 1956 Miller gave a negative answer constructing the first exotic spheres which are smooth seven manifolds homomorphic to the standard sphere of dimension seven but with non-equivalent differentiable structures and and these are these manifolds are of this form they are sphere bundles of four plain bundles over the standard sphere of dimension four and they are boundaries of the corresponding this bundle no which I denoted here by eight which has the dimension eight and these bundles are classified by two integers because as elements of the fourth a homo homotopy group of the classifying space of SO4 okay so and then so Miller considered these manifolds and well he had to prove two things first that they are homomorphic to the sphere no and to do that he used reps sphere theorem which says that if you have a closed smooth manifold if you have a Morse function with only two critical points then the your manifold is a topological sphere is homomorphic to the to the standard sphere I just recall that a Morse function is a function which has non-degenerate critical points and well now so Miller what he did is construct explicit Morse functions for their manifolds he considered and proved that they have two critical points now the the proof is easier because Milner proved that the manifolds are a homotopic equivalent to the standard spheres and now by the generalized con gary conjecture we know that they are homomorphic to the sphere but at that time Milner didn't have this this generalized con gary conjecture now the second part is to disprove the thermomorphism not to prove to see that these manifolds are not the thermomorphic to the standard sphere to do that he used here's a group signature formula which is this formula here which relates three integral invariants which are the following sigma is the signature of the intersection form of the of this smooth manifold in general and p1 square and p2 are characteristic classes of the manifold evaluated in the in the fundamental class so we get an integer and the way Milner used this formula to disprove the thermomorphism is the following so suppose that your manifold sigma seven which is the boundary of e8 is the morphic to the standard sphere so choosing a diffe morphism we can paste a disc in this manifold e8 to obtain a smooth closed eight manifold m no which is the union of e8 with this disc and we can compute the signature and p1 square and check that the quotient given by this 45 sigma plus p1 squared over seven is not an integer but here's a blue signature formula says that this should be p2 which is an integer so this is a contradiction and to the fact that we assume that sigma seven is diffe morphic to the standard sphere so in this way he proved that they cannot be diffe morphic okay so now we introduce another kind of spheres which are called Kerber's spheres for constructing them you consider the tangent disc bundle of sn and we call w which is of dimension two n is the is a manifold obtained by plumbing two copies of this tangent disc bundle a plumbing of two bundles is well illustrated with this picture and of course in the following way so if you take a disc bundle of over sn where the fibers are discs of dimension n you take a point in the sphere and take a neighborhood is this also a disc of dimension n and well in this neighborhood the bundle is trivial so you have this the bundle over this neighborhood is the disc of dimension n times the disc of dimension n and what you do is to paste to identify the the the the base of this product in one bundle with the fiber of the of the other bundle so in this cross way so what you get is something which has has corners but you can smooth these corners and get a differentiable manifold with boundary and we call the boundary k which is a smooth two n minus one manifold and is homomorphic to the to the standard sphere and we call it the Kerber's sphere because Kerber did this construction and in the case for n equal five the Kerber's sphere is an exotic sphere and if we take this manifold w which is the plumbing and and and we glue the with the cone over k nine Kerber obtained a manifold of dimension 10 which does not admit any differential structure and this was a very remarkable theorem because it has a pl structure a piecewise linear structure but she proved that you it cannot have any differentiable structure in fact it does not have the homotopy type of any differential manifold okay and to prove this she showed that the boundary was an exotic is an exotic sphere okay so now if we consider the equivalence classes of smooth structures on the sphere it has a structure of a monoid because if we consider the operation of connected some the connected some is illustrated in this picture you have two manifolds you remove two open neighborhoods to open bolts you the boundary are two spheres and identifying the spheres without the thermomorphism you get a new manifold so and with this operation in this in this in the spheres the equivalence classes of smooth structures of the spheres you get an operation and the the standard sphere is the identity element and a and well we have to record what is a h-covered ism a cover ism is a cover ism between two manifolds m and n is an h-covered ism if the inclusions of m into w and n into w are homotopy equivalences and we say that these manifolds are h-covered so there is a theorem very important bias male which says that two homotopy n spheres when n is different from three and four are h-covered and if and only if they are diffeomorphic so h-covered ism is a nice characterization of diffeomorphic homotopy n spheres so this with this theorem we can study this monoid of differentiable structures because if n is different from four this implies that the this monoid of differentiable structures on the spheres is isomorphic to the monoid of h-coverism classes of homotopy n spheres well it's male theorem didn't work for n4 and 3 but for the case n equal 3 it follows by Perlman's proof of Poincaré conjecture okay and the only missing dimension which is four is still is open there is nothing known about this structure of s4 now this monoid theta n was studied by Kerber and Milner and they prove that this is actually a group a finite abelian group and it contains a precursor group no they denoted by BP of those homotopy spheres that bound a parallelizable manifold that means a manifold with trivial tangent bundle that is why they call it BP boundary of parallelizable and they prove that is a finite cyclic group a finite index in theta n and they found the structure of this subgroup and if n is even the the group the subgroup is trivial it has order one or two if n is congruent with one modulo four and is generated by these Kerber spheres and actually is a it has order two that means the Kerber sphere is exotic if n is congruent with one modulo eight and for n modulo three n congruent with three modulo four we have that n plus one is of the multiple of four for m for some m bigger than one and its order grows more than exponentially known this formula with this bm is the Bernoulli numbers and for instance we have here the orders of some of the subgroups for dimension seven we have 28 differentiable structures for dimension 11 992 15 is 8128 and 19 is 130 816 so it's growing a lot okay so so this is what Milnor and Kerber did they found the structure of this subgroup and now well this basically well the classification of these exotic spheres and now the the second part which is has to do with singularities know how this interaction with singularities okay so the link of a singularity we take a complex analytic variety of complex dimension n in cn with isolated singular point at b so we remove p we get a complex and manifold and we have a basic proposition in singularities which is called the conic structure that there exists a small positive number such that every sphere centered in p of radius less than epsilon meets b star transversely that means that the intersection of b and the sphere is as a manifold and which is called the link of the singularity and this is more smooth real analytic manifold of dimension 2n minus one okay so this manifold this link is an invariant of the singularity and it is proved that it is independent of the radius we take of the sphere if is smaller than epsilon and well this is a the common picture in singularities now you have a isolated singularity and the intersection with the sphere is is the link and for n equal one well this link is a union of circles one for each branch of the of the complex curve and it's a not or or link in in the sphere of dimension three and this gives this very rich and nice interaction between singularities and not theory for dimension two in 1961 month for prove that if b has a normal singularity at b then the link is never simply connected so the fundamental group of the link is not trivial hence it cannot have the monopetype of a three-dimensional sphere later briskorn in 1966 proved that if b is given by this equation the first variable complex variable to the third power and the rest to the square when n is odd and bigger than three it is easy to see that it has an isolated singularity and he proved that the link is homeomorphic to the standard sphere of dimension 2n minus one and so it was an interesting result because this proved that in dimensions higher than two there is not analogous to month force theorem you know that he proved that you can have links of singularities with trivial fundamental group in fact homeomorphic to the standard sphere so this result motivated the search of exotic spheres in the links of complex singularities because when miller found the first so these spheres were these these bundles but here they appear naturally as links of singularities so the if you take a homeomorphic function from cn plus one to c which sends the origin to zero with an isolated critical point and you define the pretty much of zero which is the this complex variety and take the link of the singularity the main question is can we know when the link is a homotopy sphere and if so can we determine which element of t time represents okay so at the time when briskorn proved a hysteria he was a student of hirssebruck and also jenny was a student of hirssebruck and both the students told the results to hirssebruck and hirssebruck combined them to prove the following that if you take this this polynomial the link well it has an isolated singularity and the link is a homotopy sphere and in particular when you have these numbers three two two two is the nine-dimensional exoteric carrier sphere so here's a group presented the the first examples of exotic spheres as links of singularities and also inspired by the result by of briskorn milnor in a letter to nash considered in more generality singularities of hypersurfaces of this form you know some of the complex variables to some power bigger than two and he made a conjecture of which of them have links spheres you know as links and on the other hand FAM who was motivated by applications to the theory of elementary particles studied the same polynomials but but no equal to zero to equal to one which these are smooth complex manifolds and he computed its homotopy type the intersection form and the monodromy and these computations by FAM were the ingredients that allow briskorn to prove milnor's conjecture no so briskorn was able to prove the conjecture milnor did about which of these polynomials have links as spheres and now this with the polynomials of this form are calling singularity theory FAM briskorn polynomials because but this this results and well one example of the things that briskorn proved is the following remarkable theorem which says that every exotic sphere of dimension m which is of the form 20-1 bigger than 6 that bounds a parallelizable manifold is the link of some hypersurface singularity of some FAM briskorn polynomial for some appropriate exponents and for instance is if you take this FAM briskorn polynomial with k bigger or equal than 1 and m be equal or bigger than 2 we have a 4m minus 1 sphere which bounds a parallelizable manifold and when m equals 2 and k is from 1 to 28 we get the 28 different the FAM classes of spheres 7 spheres and from m equal to 3 and k from 1 to 992 we get also all the FAM classes of 11 spheres so then this says that you can realize any exotic sphere in this subgroup BP as a link of a singularity and well Milner consider the general question not not taking the polynomial a particular kind of polynomial but he said well let's take any any holomorphic function with isolated critical point and let's see when the link is a sphere is homomorphic to a sphere and to prove this to answer this question in this full generality a working gradient was Milner's vibration theorem which I state here so when you have F and you define the again the and the analytic variety b as the pretty much of zero and you take the link this map which I call phi which is f over the modulo of f no from the sphere minus the link to the to the circle Milner proved that is a smooth fiber bundle and and well he proved that this fiber bundle and he proved several facts about this vibration the first one is that the fiber is parallelizable so if you take also remember the the the bundle is over the circle okay so I'm going to parametrize the points in the circle by angles for titla then a fiber over theta f theta of phi is the homomorphic to the portion of the pretty much of a regular value with with argument theta contained in the ball bounded by the by the sphere which I took by to to get the link so the the picture is like this you have well the the singular variety not with with with the isolated singularity and you take a fiber of the Milner vibration and you take a regular value in the pretty much which is a non-singular fiber and Milner proved that this portion of the non-singular fiber which is inside the ball is the homomorphic to the fiber of the vibration and what he did is to construct some vector field which takes the homomorphically one into the other okay and with this well Milner made the following observations that the normal bundle of this non-singular fiber is trivial because it's the inverse image of a regular value so we have that the tangent bundle is established trivial that that means that is established parallelizable that means that if we sum the the normal the normal bundle which is trivial we get a trivial bundle and in this case when you have an empty boundary this established parallelizable implies parallelizable so we have that the link of every complex hypersurface isolated singularity bounds the fiber f t which are parallelizable manifolds so the point is to know when the link is a homotopy sphere and and when this happened which element it represents in this subgroup bp because well the links are boundaries of parallelizable manifolds okay so in this way we know that if the link is a sphere it will be an element in bp also Milner proved that the fiber f t has the homotopy type of a bouquet of spheres of dimension n remember the fiber has dimension two n so this is the the middle dimension and the number of spheres in this bouquet is strictly positive unless b has no singularity and this number of a sphere is now called the Milner number of the of the map or of the singularity and is a important invariant of the singularity no in singularity theory and also well Milner computed the homology of the link the link of every isolated hypersurface singularity is n minus two connected for n bigger than two the link is simply connected and by the isomorphism the homology is zero in dimensions one to two to n minus two and we have that the link is orientable and by point of duality also the homology vanishes in dimensions n plus i when i goes from one to n minus two so the only possibility to have non-zero groups are dimensions n and n minus one of course also in dimension zero and the top dimension which is the integers no but in the middle dimensions is just the possibilities to have non-zero groups is n and n minus one so if h n minus one vanishes then h n also vanishes by duality no and in this case we will have a homology sphere if n is bigger or equal than two then the link is simply connected no so in this case we it will satisfy the the the generalized Poincaré conjecture no we will have a hom a homology sphere with with three fundamental group so that the will imply that the link is homomorphic to the standard sphere so and so to have this you would you we just need to decide when this group h n minus one vanishes no and to to characterize when this group is zero is where Milner used his vibration theorem and he did the following well first we have to recall what is the monodromy of the Milner vibration so we have the Milner vibration no which goes recall from the sphere minus the link to the circle and this monodromy is a first return map what do i mean with this well take a fiber of the vibration and consider the unitangent vector field in the in the circle here and since this vibration is a summation you can lift it to a vector field in the sphere minus the link actually this vector field is the one we you used to prove the local triviality of the bundle and the flow of this vector field well gives a one parameter family of the morphisms which when we fix t the fiber f theta is sent to the fiber f in a plastic so if we call f the fiber of zero which of course coincides with the fiber of two pi we call the geometric monodromy the map at time to pi which sends f to f itself and we call the monodromy the induced isomorphism in the homology group hn well of course we can induce it in any dimension but as i said before the fiber has the homotopy type of a bouquet of spheres so the only non-zero group is the middle dimension and and now we define the basic polynomial of the monodromy which is this delta t which is just the determinant of the monodromy minus t times the identity and associated to the miller vibration one has a sequence in this way no which relates hn with itself using this monodromy minus the entity and then goes to the homology of the sphere minus the link and from this a sequence we can see that if this group vanishes if and only if this this map is an isomorphism and well this isomorphism can be given by a integer matrix integer square matrix and is an isomorphism if and only if is the determinant is plus or minus one so by alexander duality we have that this group hn of the sphere minus the link is isomorphic to the co-homology group of the of the link the mth homology co-homology group of the link and by Poincaré duality this is isomorphic to hn minus one of the link which is what we wanted to to characterize when it's zero so we get Milner theorem know that for n different than two the manifold given by the link of the singularity is a topological sphere if and only if this characteristic polynomial evaluated at one which is the determinant of the monodromy minus identity is plus or minus one so so this answers the question of when the link of a singularity is a homomorphic to the standard sphere of course is a differential manifold because it's the the transverse intersection of the sphere with the the variety given by the the function and now also what we would like to know which differential structure uh corresponds in the subgroup bp so uh well we saw that the the link bounds the one minus one connected parallelizable manifold f which is the fiber of the middle vibration and the diffeomorphism class in this subgroup bp uh is determined by the signature of the intersection form if n is even or by the Kerberian variant which takes values in set two if n is so well i haven't mentioned the the Kerberian variant but uh there is a characterization of the Kerberian variant by Levin that when n is so Levin proved that this Kerberian variant is given in the following way is zero if this characteristic polynomial of the monodromy evaluated to minus one is congruent to plus or minus one modulo eight and the Kerberian variant is one if this characteristic polynomial evaluated to minus one is congruent with plus or minus three modulo eight and uh so when the Kerberian variant is one it will means that the link is a exotic sphere and here uh for an even well uh i said that this characterized by the signature of the intersection form of course the the signature can be any integer but we mentioned that this group bp is a finite cyclic group uh because there is a theorem by Kerberian Milner that if the signature of two of these manifolds are congruent to uh with uh modulus of certain number then the they are diffeomorphic you know that's why we get as a finite cyclic group and the order of this subgroup is the one given by this Bernoulli numbers i mentioned at the beginning okay so then well Milner proved this this theorem to characterize when the links are spheres but then later this Milner vibration theorem has been used in singularity for many things and to to situate Milner vibration theorem in singularities i quote the there is an article by José Seade titled on Milner's vibration theorem and offering after 50 years and Seade says the following Milner vibration theorem is a landmark in singularity theory it allowed to deepen the study of the geometry and topology of analytic maps near their critical points after 50 years this has become a whole area of research on its own with a vast literature plenty of different viewpoints a large progeny and connections with many other branches of mathematics for instance by Milner vibration theorem the links of complex plane curves are fiber knots or links and i said this is there is a very rich and interesting relationship between singularities and not theory also if we take the Milner vibration together with the link this endows the the sphere with an open book structure and these get connections with contact geometry also if we take a family of complex hypersurfaces of dimension different from two and with isolated critical point the invariance of this Milner number which is a number of in the bouquet of spheres which are the monopetype of the fiber of the Milner vibration the the the invariance of this Milner number implies the invariance of the topological type of the singularity so it characterizes the topological type of singularity at the case of dimension two is still open and is a famous conjecture by Lea Romanuja and there are many other examples but this is another story that maybe i can tell you another time thank you very much thanks a lot for the talk so now i'm going to ask if you have any questions you can either type your question in the chat or any question to stop sharing because in order to be able to see the chat okay so any questions so if not i'm going to start so i would like to ask how can you use Milner vibration and the same like the the type of the singular fibers to study the the Tecohomore view of a whole variety so if you have a variety which is a vibration and you have some critical values and some singular fibers that somehow you can give in terms of Milner vibrations how can you study the the the Tecohomore view or something like that that will be more or less my question well some important when you have the vibration which is a well invariant of the singularity what you as an invariant well you want to to see when this vibrations are different in order to distinguish singularities and an important thing is the geometry and the topology of of the vibration of the Milner fiber and in the case of solid singularity well it's simple no because you have this a homology which is the bouquet of spheres which is just concentrated in one dimension but of course this has been generalized if you you don't have isolated critical isolated singularity the homology is not that simple no you have a well you can many of the groups can be non-zero and but but but this there are some also like like the middle number they are a general generalization of the middle number which are called lay numbers and it gives you idea of how is the the different homology at different dimensions so and this is also an important invariance of the of the singularity but it's not but it's more a homology of the fiber not really of the of the of the singular variety because well these studies are local no your study locally the singularity and as at the beginning I said that there there exist smaller spheres with intersect the variety and there is a theorem of there is a conical structure so locally the the variety is a cone over the and since it's a cone it's a contractible and so you don't have a homology okay thanks and I'm going to ask if there are any other questions so if not we thank again our speaker thank you very much and we wait for you on the 22nd of October for the next session and yeah and thanks a lot for attending and I'm going to send the poll right now if you're going to answer our poll