 I'm Ben Mears. I'm a postdoc over at CISA and I was asked to give this talk on an introduction to cohomology so thanks for the opportunity to speak. So my idea for this is that it doesn't seem that there are very many algebraic topologists here in Trieste and many students come without really having very much of a background in algebraic topology and so I want this to be sort of a segue for people who are maybe uncomfortable with algebraic topology to get sort of an intuitive feel so that you can go on to read some more advanced textbooks that at first might seem very technically daunting. So one of my favorite books of all time is bought into differential forms in algebraic topology. I would highly recommend it and this talk is very heavily influenced from that. So I don't want to really assume anything aside from some basic mathematical sophistication. So what is cohomology? So for any ring R cohomology with coefficients in R is going to associate a smooth manifold X with a graded R algebra cohomology of X with coefficients in R. Am I writing big enough? Besides some basic mathematical sophistication so I'm not, instead of defining everything I'm going to give some examples and let people not necessarily if you do know what these mean then that's good if you don't then you'll see some examples so that you can figure out what they mean at least in in in most cases. So yeah let me just leave it at that and it's going to get very concrete in a moment. So what I want to do is look at the case X is CP2. So this is complex projective space so this is the set of all triples C1 Z2 Z3 in C3 minus 0 and this is subject to the equivalence relation Z1 Z2 Z3 is equivalent to multiplying each of these entries by a non-zero complex scaler. Okay so in this particular case cohomology of CP2 and I'm going to take with the ring R is going to be the real numbers so CP2 cohomology with coefficients in R this is going to be a three-dimensional vector space and it's going to be spanned so so this is a grading this dot is basically a slot to indicate the grading so the zero graded piece H0 this is going to be the span of 1 H2 is going to be the span of some element I'm going to call H little H and H3 is going to be the span of H squared thank you H4 and all of the other graded pieces are going to be zero so this obviously has the ring structure H you have H times H equals H squared and H cubed is equal to zero so degree is going to be additive under multiplication so you multiply two things of degree two and you get something of degree four and so on and the other very important property is that cohomology is a functor so that means that if F is a smooth map between smooth manifolds X and Y this is going to induce a map F star going in the opposite direction of cohomology and identity maps are going to go to identity maps and compositions are going to go to compositions so an important simple consequence of functoriality is that diffeomorphisms are going to induce ring isomorphisms so so let's say Phi or I was calling it F so if F is a diffeomorphism yeah yeah the arrow is yes so the pullback goes in the opposite direction so if F is a diffeomorphism then F star is going to be a ring isomorphism so this is a simple consequence of functoriality because F is a diffeomorphism let's break this down this means that there exists a map F inverse such that F composed with F inverse is the identity on Y and F inverse composed with F is the identity on X and then you go to the ring isomorphism side or the ring side and this translates into inverse star composed with F star is the identity on H of Y so composition the order of compositions is also going to get reversed and F star composed with F inverse star is the identity on co homology on X okay so what else do we have so let me think of a simple geometric example so let's say we have a surface of genus G so this looks like this in the case G equals three if we do a reflection reflect about some point we can reflect about some plane we can reverse the orientation on this surface now CP2 has this very interesting property that the orientation that it has cannot be reversed with a diffeomorphism and this is going to follow from co homology and I'm going to explain how this happens it's essentially a consequence of this ring structure and functoriality so the point is co homology is a very useful tool because if you want to study diffeomorphisms diffeomorphisms are extremely complicated there they form this infinite dimensional space of maps but then if you apply a functor like co homology that throws away a lot of information but if you're lucky and the relevant information remains after you've applied the functor then you're working with a much much simpler object just a simple ring which is a three-dimensional vector space over the real numbers in this case and it's very easy to analyze and we'll be able to see a contradiction with what we're trying to show so trying to prove that for CP2 has no orientation reversing diffeomorphism this would be almost impossible to show directly but using co homology is very simple so so let me give a proof up to a lot of details about co homology unfortunately I won't be able to fill in 100% of the details but this should at least give you the sense of why this is true by contradiction suppose f maps CP2 to CP2 is a diffeomorphism which reverses orientation this implies that f star the induced map on the co homology rings this is going to be isomorphism of the co homology ring okay so right so this is a good point so what does CP2 bar mean so CP2 bar has a very common meaning in geometry this is equal to CP2 with with opposite the usual orientation so it's the bar indicates orientation reversal this also is very reminiscent of complex conjugation so you can take a triple in here and look at the map which sends z1 z2 z3 to z1 bar z2 bar z3 bar this is going to be a well-defined map on CP2 and it's actually going to preserve the orientation so this is not complex conjugation and note complex conjugation preserves the orientation of CP2 are there any questions I see some confused looks okay so co homology is yeah this is giving a talk like this is somewhat impossible of a certain sense because you really have to jump in and immerse yourself to understand everything if I gave all the definitions then three hours later we would yes yes because I'm going to run into that do people know what a diffeomorphism is okay well feel free to interrupt me if I'm going to get to that so so my idea is to start with the actual example here and see how the example works in as an algebraic structure so you have this algebraic structure and the properties of the algebraic structure give you this proof and then once you see how the algebraic structure reflects the structure of the CP2 then we can go and try and understand how the structure is built up so that's my plan okay so so f star always preserves the degree so in other words f star is going to map h0 to h0 h2 to h2 and h4 to h4 so f star being a ring homomorphism so that means that you have a ring so a ring is something with a structure of addition and multiplication satisfying the normal rules so f star being a ring homomorphism means it's going to preserve it's going to send addition to addition and multiplication to multiplication so in particular so we have an element one an element h and an element h squared and I need to tell you where these can possibly go under a ring homomorphism so in h0 we have to have one being mapped to one so where can we map h to so h is going to map to something that's in the span of h so it's going to be some lambda times h for some lambda in R so this is not going to be specified it's going to be a parameter of the of of the ring homomorphism and now h squared maybe I should write this as f star of one is equal to one f star of h is equal to lambda h f star of h squared this because it's a ring homomorphism this is f star of h squared so this is lambda h squared which is lambda squared h squared why do these have to map to each other okay so these super scripts indicate the degree so I have a decomposition of my ring into pieces of different degree so degree zero degree two and degree four so whenever I multiply two things whenever I multiply two things I'm going to add the degrees so so the span of one this is the span over the real numbers so this is going to be just a copy of the ring R so in degree h0 I have just the ring R so span of h this is r times h and the span of h squared this is r times h squared so if I multiply something in h2 by something in h0 this is a scalar times some fact some multiple of h that's going to go from that that's going to end up in h2 because 0 plus 2 is 2 but if I multiply two things in h2 then that's going to end up in h4 if I multiply something in h2 times something in h4 then that has to end up in h6 and we're going to see later on that the small h is going to be represented by the scalar form on CP2 yeah it's a two form h squared is going to correspond to the volume form it is but I should continue with the proof I think okay so there's one other thing which I haven't told you yet and that is that an orientation of an n manifold is going to induce a non-zero linear map and I'm going to write this linear map with the symbol of integration over x so this is going to be very suggestive of what these objects represent this integration over x this is a linear map from hn of x to the real numbers and the opposite orientation is going to induce so integral over x bar where the bar represents the opposite orientation for x this is equal to minus the one of the usual orientation h squared is going to be twice the volume form does that make you happier but it's still h squared yeah and it is the volume form using the wedge product no the wedge square only vanishes on odd forms for even forms the wedge square can be non-zero so we're kind of having our own little discussion here so I think I should okay so if f is orientation preserving then so I want to explain how this integral map that is induced from the orientation changes under an orientation preserving diffeomorphism so so k is going to be an element in the nth cohomology and if I compare the integral map on k to the integral map on f star applied to k these are going to be the same and if f star is orientation reversing then you're going to have a minus sign I can't hear you oh yes yes thank you if the original function f then the induced map on the ring we don't have orientations on rings except for this is the corresponding notion for an orientation on a ring okay so from here we are essentially done after a quick computation because so for cp2 all we need to do is plug in k is equal to h squared so if we do this then we find that integral over x of f star of h squared this is equal to well on one hand we know that this is equal to integral over x of lambda squared f star of h on the other hand so this is using the ring homomorphism on the other hand we have that this is equal to minus sorry h squared h squared on the other hand this is equal to minus h squared by the orientation reversing so this is a linear map so I can pull out the lambda squared and what I get is that lambda squared plus one times integral of x of h squared is equal to zero so did I go too fast so pull out the lambda squared here and then bring this to the other side so I get one plus lambda squared times the integral so this means that either lambda squared plus one is equal to zero or this integral here is equal to zero now this can't happen because this is a non-zero linear map no I can take that positive number to be one so the induced map on on cohomology is if you think of integer cohomology the real cohomology is the image of integer cohomology in there and under integer cohomology that number is always one for instance right I don't have yeah yeah and in general you're going to get the degree as a multiplicative factor out there so it's always going to be an integer but for diffeomorphism it has to be plus or minus one okay so because this is a non-zero linear map and because h4 is spanned by h h squared the only way that this this integral has to be something non-zero so this implies that lambda squared plus one is equal to zero for a real number which is a contradiction so the structure of the cohomology ring is is prohibiting diffeomorphism which reverses the orientation and this is at least in my opinion the heart of what makes cohomology a very powerful tool so maybe I should reflect a little bit more on on what this is saying okay so if so in general because we can divide by the integral of h squared what what this is saying is that lambda squared is equal to the integral over x of f star of h squared over integral of x of h squared so recall that so f star is going to add a factor of h squared here which you can pull out and then the denominator just cancels so this is lambda squared written in terms of this integral here and this is equal to minus one if if f is orientation reversing and plus one if f is orientation preserving so the only possibilities are lambda is plus or minus one and f preserves orientation so for a diffeomorphism what you can do basically in in terms of trying to swap the orientation you can try and swap the sign on h but for h squared you can't change the sign if you're doing this over the complex numbers then this argument would break down but because we're doing this over the real numbers and we can do this over the real numbers because I can choose this ring to be whatever I want then we have this contradiction of not being able to flip the sign so what this is saying is that as orientation or oriented manifolds cp2 and cp2 bar are distinct manifolds up to orientation preserving diffeomorphisms so oh sorry I are people not able to see this so as oriented manifold cp2 and cp2 bar are distinct manifolds up to orientation preserving diffeomorphisms so strangely enough cp2 is not the simplest example of such a manifold it's a little bit difficult to see this because this case is so so degenerate but the zero-dimensional manifold consisting of a single point with some chosen orientation which corresponds to either plus one or minus one so positively oriented point this has no orientation reversing diffeomorphism okay so I was going to sketch the details of how how this ring arises from cp2 so how do you go from cp2 to this ring structure so cp2 is a complex manifold this means that the tangent bundle of cp2 has an endomorphism J and this satisfies J squared is minus the identity and this endomorphism is what induces the complex structure now cp2 also has a special metric called the Fubini's Studi metric and associated this is a Riemannian metric on cp2 and it has a Levy-Chavita connection and you can look at the covariant derivative of this endomorphism and it turns out that for this special choice of metric this complex structure is covariantly constant it may not be the simplest way but the there there is there is no especially simple way that I can think of to express this so this is just one of many ways of of going about this this is I think the most concrete way of I mean yeah so it's so I I don't expect everyone to understand every word that I'm saying but the essence of what I'm saying doesn't require understanding every single word but but the point is that there's something called a scalar form and this is very easy to define so if G is the Fubini's Studi metric so G evaluated on two-vector on two-vectors X and Y gives the dot product of these two vectors these are two vectors in the same tangent space and I can act by the complex structure on one of these vectors and this is going to be anti-symmetric and omega of XY is defined to be this anti-symmetric bilinear form on the tangent bundle and so this H is going to be associated with omega so at a single point omega is going to be it's going to look something like I actually I think I should leave it at that for now and omega squared so so H squared is going to be associated with the volume form okay and even more simply one in H zero this is going to be the constant function one okay this obviously isn't a complete explanation and then finally this linear map which is associated with integration this is going to correspond to integration of the volume form on on the man on on CP2 okay so we have these three objects I haven't really explained how you get them but you can ask how do you know that the cohomology ring doesn't contain anything else and this again I can't really give a complete explanation but there's something called the Meyer v. Atourist sequence so Meyer v. Atourist this if you decompose a manifold x into the union of two sets a and b this relates the cohomology of x with cohomologies b and the intersection of a and b so one thing you hear in topology quite often is somebody computes the cohomology of something and in my view it's a little bit of a misnomer because you don't actually plug cohomology into a formula and get an answer out after after doing some simple computations but you have some relation between the cohomologies of simpler spaces and spaces that are built up from patching together these more complicated spaces and in practice what you do in order to compute something like this is you take the cohomology of simpler things and use that to infer the cohomology of the larger things indirectly so computation of cohomology is usually very indirect so let me give some other examples so cohomology of cpn with real coefficients this is going to be the ring r with a variable h subject to the relation h to the n plus 1 is equal to 0 so in the case of cp2 I have the ring that's generated by h and subject to the relation h cubed is equal to zero so I have one h h squared and nothing more so in cpn you have one h h squared h cubed all the way up to hn so it's the span of one h h squared hn it's possible to plug in other manifolds so cohomology of rp2 let's say rp2n this is the same as the cohomology of a point which is just the span of one so it's a very boring cohomology ring so cohomology sometimes throws away too much information and you lose all of the structure but there are ways around this so here I'm using real coefficients if I were to look at cohomology with integer coefficients you might think well the integers are smaller than the reals so I should get some smaller answer but it turns out that you get something bigger so you get integers with h modulo so 2h is equal to zero so h is a torsion element 2h is equal to zero and h to the n plus 1 is equal to zero oh and I should have said that the degree of h is equal to 2 the degree of h here is also equal to 2 and so in terms of the vector space well a vector space over a ring that's not a field is called a module it's the same idea though so in this case we have it's the this is z in degree 0 0 in degree 1 z mod 2z in degree 2 and it's going to alternate between z mod 2z up until z mod 2z in degree n and then zero after so the difference between coefficients in the integers and coefficients in the reals is that z mod 2z this is something non-trivial consisting of zero one r mod 2r well that's zero as a final example I want to consider the two torus so the cohomology of s1 times s1 with real coefficients this is going to be a ring with two variables a and b subject to the relations a squared is equal to zero b squared is equal to zero and a times b is equal to minus b times a so the ring is not commutative it's actually super commutative so odd cohomology acts like fermions and even cohomology acts like bosons so the product satisfies the rule x times y is equal to minus one times the degree of x times the degree of y times y times x so x times y equals y times x unless both x and y are odd degree in which case the sign flips yeah so I prepared a whole lot more than this so I don't know what people want to do so I guess I can mention what my plan is and depending on if people have the will or energy to actually see this I can do it or just skip this but I was going to explain how to the simplest method of defining cohomology just dirham cohomology so I was going to explain how to get this from first principles so looking at locally constant functions how you're naturally led to an anti-symmetric product and differential forms and the exterior derivative and how that naturally leads to cohomology so I'm sorry it took so long and I didn't really seem to get very far but yeah yeah so probably it makes the most sense at this point to take a break and let people decompress a little bit and then if people are still around then I can give some I can basically motivate the definition of dirham cohomology in a really slick way