 Today we will do some combinatorial study of simplicial complexes, of course there is some topology there but mostly it is the relation between you know where something is incident and so on, neighborhood and such things which are all studied by the combinatorial relation, combinatorial relation is precisely which phase is where, that kind of information. These things are useful if we are doing many more combinatorial problems and also triangulation of manifolds and various things, polyadipology and so on. Within this course we are not going to use this one, so there is less emphasis on this one. There is one reason why I put it at the end of study of simplicial complexes, just because we are not going to study much of it does not mean that this is less important. Okay, so for this study to begin with you can just concentrate on finite simplicial complexes, to get a hold on what is going on okay, but all the definitions and all the results except when I stated clearly they are valid for any simplicial complex. But to get your ideas clear you just work on some finite simplicial complex what is happening okay, so take a finite simplicial complex, f be any simplex in k, recall we have already defined this open simplex to be a subset of mod k which consists of all alpha and mod k with alpha v not equal to 0 for every point v inside f and conversely. That is alpha is not equal to it should not leave v is inside f. So roughly simply if inside mod f which is all alpha v such that the support is contained inside f okay, so this set is actually the interior of that. It may not be the interior of mod f inside mod k, this will be an open set and interior of something provided f is a maximal simplex. So all this we have seen in general this open simplex need not be an open subset of mod k. In fact this is open if and only if f is a maximal simplex. Note that mod f is equal to open f if and only if f is a single term all these things we have seen but I am just you know drawing your point again drawing your attention to these things. An important thing to note is that open simplex is of in any simplex complex or my partition of mod k. This is the fact we have used also okay. Now let us do this little more. Now take l to be a sub complex of k and f be inside l then look at all simplex is h that is that h union f is still a simplex of l it must be simplex of l not of k okay inside l. h union f should be a simplex then you put h in this one in this collection in particular f union f is also there right. So f will be also there and subset of f will be also there okay. Once something is in place all subsets of h will be also there not only that even subsets of f are also there. So this thing forms a sub complex of l I have just told you why if h prime is a sub of h then h prime union f will be also simplex because l is a sub complex. So this has a name this is called star of f and we denote it by star of f in l because these things are they can inside l okay in particular if l is the whole of k then we can take st in a suffix k or you can just drop it because the ambient simplex complex k is fixed it may not any need to write it. The star of f if you take various different sub complexes that will be different because then it will be only those things which are inside that sub complex is taken that is all. The modulus of star f which is the underlying topological space is naturally identified with the subspace of mod k okay this is true for any sub complex after all right and this is nothing but the union of all closed simplex is mod g in k that contain f as a g if contains f and it is a sub complex of k it will be inside star by definition okay this subspace is referred to as closed star of f okay it is called closed star of f in k it is the closed subspace of mod k containing mod f okay it is quite large of course not a smaller one the mod f itself is closed but this will be another large closed subset is called closed star okay there will be an open star also we will define that so on in contrast there is a you see this is capital st here small st small st of f defined as the union of all open simplex is okay g open such that g is a simplex of k containing f this is called the open star of f in k clearly this is an open subspace of mod k because you have taken all open subsets larger than that also so it will be open subsets of mod k and it contains this open f so this open f is dangerous you have to be very careful with this for example this open f does not contain all points of mod f okay it may not contain it actually does not contain mod f in general okay maybe this will contain only if f is a single data for example inside a triangle empty triangle okay only three sides are there take one of the sides as f then what is open f open f is open interval of that of that except the two points are gone now what is open star of f that you have to take of course you have to take this cell and then you can take a triangle and take the open triangle so this will be the open triangle union the open simplex on on one side whichever one you have taken okay so that will be the open subspace of it's called open star of containing this one open open star of f where f was one of the edges okay so this example whatever i given must make it clear what is happening here inside the closed triangle the full triangle okay this open star will be an open subset now i define another concept called link okay so this topic is called links and stars so another concept called link the link of f in l so l is a complex f is some intelligence on of f in l is defined to be the sub complex of star of f remember what is star of f it contains all g such that g union f is a simplex inside l you look at that but only take those consisting of simplices which are disjoint from f say some g is there g intersection f is empty but g union f is inside l because it must be in star l so if it is far away then that will not be taken the collection of all such things will be called link of l link of f inside l once again if l is the whole of k we just drop this suffix and there is a link of f the underlying topological space is naturally it is subspace of mod the closed star of f and of course is subspace of mod k also and what is this much union of all closed simplex mod g such that inside this one which are disjoint from mod f okay so i will give you examples of links also first by just purely commentarily observe what happens to dimension of star f dimension of star f is actually equal to dimension of link f plus dimension of f plus 1 why look at any simplex here okay what is the definition it is g union f inside that one which is simplex right and what is simplex here it is g union f but g intersection f must be empty so starting with a simplex here you write it as g union f where g intersection f is empty you can write any union as disjoint union right because f is upset then that g which is disjoint from f will be as link will be an element here what is the dimension of g disjoint union f number of elements in g plus number of elements in f right minus 1 dimension of each of them is number of elements minus 1 therefore you get this plus dimension of plus plus this f is kept fixed the element which comes here is disjoint from f such that union is here so this formula is obvious okay so that is what i mean by what we are doing is purely combinatorial no topology here but this is a only comment or but now i will i will tell you a stronger result which topology here okay or geometry and that is the result here take any simplex complex in any phase f okay we have the star of f i am not writing the sub sub complex here l but if you write that that will be also true because this is true for every simplex complex okay i can put an l here then it will be all true star of f is equal to link of f what is this one the joint join with f what is this f f is a simplex complex itself it's like a delta n if n is the dimension of f all all subsets of that are also taken that is the meaning of f okay as a simplex complex to take the the star a link join of these to be that this one so this is again combinatorial but what is the meaning of this when you take when you pass through the geometric realization geometricalization of star of f is homeomorphic to the joint namely the geometricalization of link star with this is joint geometricalization of f this part we have already seen for any k 1 and k 2 you first take the joint and then take the underlying topology in mod k mod of that is same thing as first take the mod and then take the joint okay so only this part you have to check and this is precisely the kind of thing i did here for each f here this is disjoint from every element here is disjoint from that but the union is in the link in the joint and you have to put it here by definition and conversely starting with anything you can rewrite it as f disjoint union g that g will be here that's the b okay it may happen that it is not the full f it may be some part of it right so that's what you have to be careful about this all right so suppose i take a sub subset here by very definition all subsets of that will have to be also there because this is a simple complex so after taking this one you can take the subset also there now here is okay in particular yeah important so i saw this one already but here is i have given the explanation here so this is our standard straightforward we have seen it several times now as a consequence in a particular take any point inside the open simplex then we have link of f okay start with the boundary complex of f okay instead of taking the full thing is contained inside link of f start with the whole thing minus x right x is a single point here so you throw that that point is not there anyway in the whole thing also because we have taken on the boundary here but this itself is the full star minus point okay so this much is obvious but what happens is this is a deformation retract of the star of f minus x is subspace here is a deformation retract what does it mean that there is a homotopy of the identity map of this one okay at the end it is a retract it's a map on to this part which is identity on the boundary on this part okay so here is the proof fix a homomorphism from the boundary complex star singlet and x this is a core in the whole space we know this one you fix some linear homomorphism t times explosion minus t times etc etc fix a homomorphism then look at link of f star bf star x so this i already know i am taking again the join with this one link of f this is homomorphic to mod f right boundary of star x link of x star i am putting bracket here that's all by associativity first bracket was here now the bracket is here but once this is f i can put f here link of f star f we have seen that it is star okay so this is one way of looking at it on the other hand we know that for any space y the base y cross 0 of the cone is a deformation retract of the entire cone minus the apex okay i use that take y here to be link of f star bf this is the base and that is the cone okay so conclusion is that the star of f minus this point will deform retract on to this part okay so we have developed these things earlier in abstract so i am applying it in the special case here now k base impartial complex f and g be any two disjoint phases of a phase f union g the start with the phase and then write it as disjoint union that's all i am talking about in k then look at a link of f in star of g okay look at the star of g it contains f because g composite g union f is a simplex therefore this makes sense so f is a f is some simplex in the sub complex this is my taking star of g is like a sub complex l okay the link inside this one of f is star of g in the link of f now you can see why the title of this section was links and stars link of star is star of the link okay of course f and g get interchange f g g f okay purely commentary lizard there is not apology here you see it's just commentary lizard you have to check left hand side equal right hand side take a simplex here show that it is here and vice versa okay so okay first observe that f intersection g is empty this started assumption and the union is in k therefore this g okay is in the link of f and this f is in the star of g okay therefore under the given hypothesis nearly f and g etc have disjoint what we have is suppose l is sub complex l is not a sub complex l is a phase of star of g in the link of f i am starting from here this just means and it's if it only if l is first of all in the link of f and l union g must be in the link of f that is the definition of star right what gets is the same thing as saying l union f is a simplex in k l intersection f must be empty because it is in the link l union g union f must be inside k and l union g that is something wrong here l union g must be also inside k okay so this is same thing as l union g inside k l intersection f is empty l union g union k inside k i am just rewriting this one so nothing is lost here but this is the same thing as l inside star of g by the definition l intersection f is empty l union f is in star of g this is the definition of because l union f union g is inside k so this is in star of g okay so notice that to prove something inside link of f and link of g i am going all the way to k okay because these things were taken inside k link of f is inside k similarly star of g is also inside k so that's why i am going inside k so this just means that l itself is in the link of star of g this is in star of g and it is intersection with f is empty therefore this is entirely gone and this is all reversible all if any all implications are reversible that's what we are saying okay this result was used by munkrae in deriving something about some result which algebraics were interested in i am not able to state it here in this particular case to prove that result this is called reissner's condition for a topological spaced way homology sphere so munkrae's proved that that the condition is true for whatever the kind of thing there is true and that became key result for reissner to prove his result in commentaries which was in turn used by stanley to to prove a big theorem of upper bound conjecture and form okay so there are history here so that is one of the ideas why i just spent some time on these links in stars the only thing is unless you are familiar with little more familiar with them even the definition you may forget okay so to get familiar you can star you can solve a few exercises here which i have mentioned okay for example show that star of g inside star of f okay star of f intersection star of g both taken inside k the only thing you have to assume that is f union g is a simplex then something happens you can just prove or disprove it then you will know you will you would have worked out you would have known whether you have understood these things correctly okay so that is all for today thank you