complex geometry - How can I see the projection $pi:H^1(X,mathcal{O}_X)rightarrow Pic^{0}(X)$ in terms of holomorphic structures on $Xtimesmathbb{C}$?

Hi, as the title says I'm looking for a way to see the projection $pi:H^1(X,mathcal{O}_X)rightarrow operatorname{Pic}^{0}(X)$ in terms of holomorphic structures on $Xtimesmathbb{C}$. ($X$ is a compact complex manifold and $operatorname{Pic}^0(X)$ is the kernel of $c_1:H^1(X,mathcal{O}_X^{ * })rightarrow H^2(X,mathbb{Z})$ in the long exact sequence induced by the exponential sequence). Since
$$H^1(X,mathcal{O}_X) simeq H_{overline{partial}}^{0,1}$$
by the Dolbeault isomorphism, I take
$[c]in H^1(X,mathcal{O}_X)$,
so I take the corresponding $[gamma]in H_{overline{partial}}^{0,1}$ and a representative $gamma$ of the class $[gamma]$. My wrong thought was that $pi([c])=(Xtimesmathbb{C},overline{partial}+gamma)$, i.e. to associate to $[c]$ the trivial bundle with the "holomorphic structure" $overline{partial}+gamma$, but it is not a holomorphic structure unless $gammawedgegamma=0$! So how can I see explicitly (if it is possible) the map $pi$ in terms of holomorphic structures on the trivial line bundle?

Thank you in advance.

Set theory and alternative foundations

This was going to be a comment to Joel David Hamkins's answer on geometry, but it didn't fit.

+1 This is one of the most clear-minded things I have read on MO. It does not make a mockery of Foundations and still says something non-obvious.

I'm very skeptical of all this business with category theory being a foundation for mathematics. First, whenever anyone talks about it, it always seems to be somebody else's work. It's become something of a meme that "Bill Lawvere has proposals to provide foundations for math through category theory", but we don't ever see details provided.

Second, are we really to understand that we are going to add small integers with arrows and diagrams? Draw circles and lines, and say that the latter meet at most once? I think people work in trans-Euclidean hyperschemes of infinite type so much, they forget that math includes these things.

As Wittgenstein remarks in the Investigations, just because you can express A in terms of B, it does not mean that B actually underlies A.

ag.algebraic geometry - Cohomology of Zariski neighborhoods

Do there exist smooth compact (=complete) connected complex algebraic varieties $Xsubset Y$ and a Zariski neighborhood $U$ of $X$ in $Y$ such that the image of $H^{ast}(U,mathbf{Z})$ in $H^{ast}(X,mathbf{Z})$ under the restriction map is different from the image of $H^*(Y,mathbf{Z})$?

Remark: if one considers the rational cohomology instead of the integral one, the answer is no by Hodge theory: if a class on $X$ comes from a class on $U$, it is the restriction of a class of the "right" weight and hence it extends from $U$ to any smooth compactification, in particular to $Y$. In this argument it is important that cohomology maps induced by regular maps of algebraic varieties are strictly compatible with the weight filtrations: if a weight $k$ class in the target is in the image, then it is the image of a weight $k$ class.

The motivation behind the above question is to understand whether or not the same holds for the integral weight filtrations, which can be defined as the Leray filtrations induced by the open embedding in a compactification as the complement of a divisor with normal crossings, see Weight filtration over the integers.

The above is in a sense the simplest possible situation when strictness may fail. I.e., if the answer to the above question is positive, this would imply that the integral cohomology mappings are not necessarily strictly compatible with the weight filtrations. If the answer is negative, then I would be very interested to know the answer to the same question with "algebraic" replaced with "complex analytic" and "Zariski neighborhood" replaced with "an open set whose complement can be blown up to a divisor with normal crossings".

ag.algebraic geometry - Why a subvariety of a variety of general type is of general type

You need to be careful about what you mean by "a general point". Usually, this means "a point in a certain Zariski open set". So in particular, this statement would say that on a surface of general type all rational and elliptic curves lie in a Zariski closed subset. This is Lang's conjecture, still open I believe (proofs were suggested about 15 yrs ago but then withdrawn).

EDIT: OK, so from the comments and the other answer it appears that it should be a "very general point of $X$", and the statement reduces to showing that if you have a morphism $f:Yto T$ with irreducible $T$ and general fiber $Y_t$, and a finite dominant morphism $pi:Yto X$, then $Y_t$ is also of general type.

The basic reason for that is very simple: if $X$ is of general type then it has lots of pluricanonical forms. You can pull them back to $Y$ (they are differential forms, after all) and get lots of pluricanonical forms on $Y$. Then you can restrict them to $Y_t$ and get lots of pluricaninical forms on $Y_t$.

For a more precise answer, I suggest you look at old papers by Kawamata, Viehweg and Kollar, search for "additivity of Kodaira dimension". There is a whole sequence of $C_{n,m}$ conjectures about the Kodaira dimension of a fibration $Y$ in terms of Kodaira dimensions of $T$ and $Y_t$. Some of them are proved, some are still open.

(Note: general type means "maximal Kodaira dimension", i.e. equal to the dimension of the variety.)

gr.group theory - Can all terms of the Johnson filtration be hom-mapped onto the same nontrival group?

The answer seems to be affirmative. We use the idea of Henry Wilton that the image might be taken as an alternating group $A_q$, a simple one (see his comment above). Let $K=mathcal A(1).$ Then

$mathcal A(m) ge [K,K,ldots,K]=[..[K,K],..,K]qquad (mquad times) qquad (*)$

Take a nontrivial $alpha in [K,K]$ and a surjective homomorphism $Delta: mathrm{Aut}(F_n) to A_q$ which doesn't vanish at $alpha$.

Then
$$
A_q =mathrm{NormalClosure}(Delta(alpha))=Delta([K,K])=Delta(K).
$$
It follows that
$$
Delta( [K,K,ldots,K])=A_q
$$
and by $(*)$ $Delta( mathcal A(m))=A_q $ for every $m ge 1.$

gt.geometric topology - Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and only if the links can be related by a sequence of Kirby moves and isotopies. This is pretty similar to Reidemeister's theorem, which says that two link diagrams correspond to isotopic links if and only if they can be related by a sequence of plane isotopies and Reidemeister moves.

Note however that Kirby moves, as opposed to the Reidemeister moves, are not local: the second Kirby move involves changing the diagram in the neighborhood of a whole component of the link. In "On Kirby's calculus", Topology 18, 1-15, 1979 Fenn and Rourke gave an alternative version of Kirby's calculus. In their approach there is a countable family of allowed transformations, each of which looks as follows: replace a $pm 1$ framed circle around $ngeq 0$ parallel strands with the twisted strands (clockwise or counterclockwise, depending on the framing of the circle) and no circle. Note that this time the parts of the diagrams that one is allowed to change look very similar (it's only the number of strands that varies), but still there are countably many of them.

I would like to ask if this is the best one can do. In other words, can there be a finite set of local moves for the Kirby calculus? To be more precise, is there a finite collection $A_1,ldots A_N,B_1,ldots B_N$ of framed tangle diagrams in the 2-disk such that any two framed link diagrams that give homeomorphic manifolds are related by a sequence of isotopies and moves of the form "if the intersection of the diagram with a disk is isotopic to $A_i$, then replace it with $B_i$"?

I vaguely remember having heard that the answer to this question is no, but I do not remember the details.

matrices - Expected determinant of a random NxN matrix.

As everyone above has pointed out, the expected value is $0$.

I expect that the original poster might have wanted to know about how big the determinant is. A good way to approach this is to compute $sqrt{E((det A)^2)}$, so there will be no cancellation.

Now, $(det A)^2$ is the sum over all pairs $v$ and $w$ of permutations in $S_n$ of
$$(-1)^{ell(v) + ell(w)} (1/2)^{2n-# { i : v(i) = w(i) }}$$

Group together pairs $(v,w)$ according to $u := w^{-1} v$. We want to compute
$$(n!) sum_{u in S_n} (-1)^{ell(u)} (1/2)^{2n-# (mbox{Fixed points of }i)}$$

This is $(n!)^2/2^{2n}$ times the coefficient of $x^n$ in
$$e^{2x-x^2/2+x^3/3 - x^4/4 + cdots} = e^x (1+x).$$

So $sqrt{E((det A)^2)}$ is
$$sqrt{(n!)^2/2^{2n} left(1/n! + 1/(n-1)! right)} = sqrt{(n+1)!}/ 2^n$$

ag.algebraic geometry - Reference for cohomology vanishing

To expand on Emerton's answer: Using the excision sequence, Cartan's result in the algebraic case boils down to showing the following: Let $R$ be a regular local ring, and $I$ and ideal of height at least $3$, then $H^i_I(R)=0$ for $ileq 2$.This follows because: $$H^i_I(R) = lim Ext^i(R/I^n,R)$$

And $I^n$, being height $3$, always contains a regular sequence of length $2$, so the $Ext^i$ vanishes for $ileq 2$ by standard result (see Bruns-Herzog Cohen Macaulay book, Proposition 1.2.10 for example). This argument extends to the case of codimension at least $n$ and vanishing of $H^{n-2}$.

Incidentally, a pretty non-trivial question is to find upper bound for the vanishing of local cohomology modules, in other words, the cohomological dimension of a subvariety $Z$. Many strong results have been obtained after SGA, by Hartshorne, Ogus, Faltings, Huneke-Lyubeznik, etc. All those references can be found in
Lyubeznik's paper (they were mentioned in the very first page) which primarily treated the vanishing of etale cohomology.

ac.commutative algebra - Decomposition of modules using computer packages

Suppose A = C[[x,y,z]]/f(x,y,z) is one of the ADE singularities, where there are finitely many indecomposables P_1,...,P_n. In analogy with character theory of finite groups, we want to set up a situation where Hom(P_i,P_j) = delta _ij. That will allow us to decompose a reflexive A-module into a direct sum of indecomposables (in the same way one decomposes a representation into a direct sum of irreducible representations).

The triangulated category underline{CM}(A) = CM(A)/A has a t-structure with heart CM(A), in which the finitely many indecomposables are the simple objects. The simples satisfy Hom^0(S_i, S_j) = delta ij. To compute Hom^0 in this category use the equation underline{Hom}(M,N) simeq Ext^2_A(M,N). (See Burban and Drozd's survey paper, especially page 46.)

So basically you just have to compute Ext^2 in the complete local ring A. Singular can do this directly.

If you don't want to download Singular, Macaulay2 can do it, although it takes some care, because Macaulay naturally works with graded modules over polynomial rings; (one has to be careful with grading shifts.) For more information on the graded case, see the papers by [Kajiura Saito and Takahashi].

soft question - Origins of names of algebraic structures

I'm not sure that I'm historically accurate, but that is how I always thought about algebraic nomenclature.

1) Group actually comes from group of substitutions. I guess that Galois could have introduced any other word, like "set" of substitutions or "flock" of transformations. Set theory was not yet established, so I guess a collection of functions could be called 'group', 'set' and so on according to the taste.

2) For field, I guess it comes from the meaning of field as "sphere", "subject", "area". It makes sense that such a word could come in talking about "solving an equation in the real field" rather than "solving an equation in the complex field". Then the concept of an abstract field could have followed.

3) Ring comes from "Zahlring", ring of numbers. This, as far as I know, is a terminology due to Dedekind. He was actually working with number rings, of the form $mathbb{Z}[alpha]$, where $alpha$ is integral over $mathbb{Z}$. So for some $n$, $alpha^n$ can be expressed in terms of lower powers of $alpha$; in some sense the components of the basis of $mathbb{Z}[alpha]$ over $mathbb{Z}$ cycle, although this is accurate only when $alpha$ is a root of unity. Hence the name ring of numbers.

4) Ideal is easy. When Dedekind realized that in a ring like $mathbb{Z}[sqrt{-5}]$ unique factorization does not hold, he searched for a substitute. He then realized could restore unique factorization allowing something more general than elements, the ideals. These are now called this way since he thought of them as "ideal elements" of the ring. useful to restore unique factorization. It is a fortunate coincidence that indeed for the rings he was working with (which are now called Dedekind rings), unique factorization for ideals actually holds.

5) Idéle has the same origin, being the contraction of the French "idéal élement", although the wording is inverted with respect to French use.

examples - Connection between bi-Hamiltonian systems and complete integrability

As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.

Two Poisson brackets
${ cdot,cdot } _{1} , { cdot , cdot } _{2}$ on a manifold $M$ are compatible if their arbitrary linear combination
$lambda { cdot , cdot } _1+mu{cdot,cdot} _2$ is also a Poisson bracket. A bi-Hamiltonian system is one which allows Hamiltonian formulations with respect to two compatible Poisson brackets. It automatically posseses a number of integrals in involution.

The definition of a complete integrability (à la Liouville-Arnol'd) is:

Hamiltonian flows and Poisson maps on a $2n$-dimensional symplectic manifold $left(M,{ cdot, cdot }_Mright)$ with $n$ (smooth real valued) functions $F _1,F _2,dots,F _n$ such that: (i) they are functionally independent (i.e. the gradients $nabla F _k$ are linearly independent everywhere on $M$) and (ii) these functions are in involution (i.e. ${F _k,F _j}=0$) are called completely integrable.

Now, I would like to understand the connections between these two notions, and because I haven't studied the theory, any answer would be helpful. I find reading papers on these subjects too technical at the moment. Specific questions I have in mind are:

Does completely integrable system always allow for a bi-Hamiltonian structure? Is every bi-Hamiltonian system completely integrable? If not, what are examples (or places where to find examples) of systems that posses one property but not the other?

I apologize for any stupid mistakes I might have made above. Feel free to edit (tagging included).

dg.differential geometry - Differentiable function germs on differentiable manifolds

Hello everyone, I was wondering if anyone knew how to prove that the map from $C^{infty}(M)$ to $xi (p)$, that is, from the infinitely differentiable functions on a manifold M to the space of (once)-differentiable function germs, where the map is associating to each f in $C^{infty}(M)$ its class in $xi (p)$ is onto.
By the way, since you ask, the reason I'm interested in this is because its a question that WAS on my final for differential topology, I've tried to work it out since then but no luck so far, this is not homework it's just curiosity now, hope its ok ill have to check the post regultaions, sorry, if not just tell me and i'll delete the question...

mirror symmetry - Higher genus closed string B-model

This is a great question I wish I understood the answer to better.
I know two vague answers, one based on derived algebraic geometry and one based on string theory.
The first answer, that Costello explained to me and I most likely misrepeat,
is the following. The B-model on a CY X as an extended TFT can be defined in terms of
DAG: we consider the worldsheet $Sigma$ as merely a topological space or simplicial set (this is a reflection of the lack of instanton corrections in the B-model), and consider the mapping space $X^Sigma$ in the DAG sense. For example for $Sigma=S^1$ this is the derived loop space (odd tangent bundle) of $X$.. In this language it's very easy to say what the theory assigns to 0- and 1-manifolds: to a point we assign coherent sheaves on $X$, to a 1-manifold cobordism we assign the functor given by push-pull of sheaves between obvious maps of mapping spaces (see e.g. the last section here). For example for $S^1$ we recover Hochschild homology of $X$. Now for 2-manifold bordisms we want to define natural operations by push-pull of functions, but for that we need a measure -- and the claim is the Calabi-Yau structure (together with the appropriate DAG version of Grothendieck-Serre duality, which Kevin said Lurie provides) gives exactly this integration...

Anyway that gives a tentative answer to your question: the B-model assigns to a surface $Sigma$ the "volume" of the mapping space $X^Sigma$, defined in terms of the CY form.
More concretely, you chop up $Sigma$ into pieces, and use the natural operations on Hochschild homology, such as trace pairing and identification with Hochschild cohomology (and hence pair-of-pants multiplication).. of course this last sentence is just saying "use the Frobenius algebra structure on what you assigned to the circle" so doesn't really address your question - the key is to interpret the volume of $X^Sigma$ correctly.

The second answer from string theory says that while genus 0 defines a Frobenius manifold you shouldn't consider other genera individually, but as a generating series -- i.e. the genus is paired with the (topological) string coupling constant, and together defines a single object, the topological string partition function, which you should try to interpret rather than term by term. (This is also the topic of Costello's paper on the partition function). BTW for genus one there is a concrete answer in terms of Ray-Singer torsion, but I don't think that extends obviously to higher genus.

As to how to interpret it, that's the topic of the famous BCOV paper - i.e. the Kodaira-Spencer theory of gravity. For one thing, the partition function is determined recursively by the holomorphic anomaly equation, though I don't understand that as "explaining" the higher genus contributions. But in any case there's a Chern-Simons type theory quantizing the deformation theory of the Calabi-Yau, built out of the Kodaira-Spencer dgla in a simple looking way, and that's what the B-model is calculating.
A very inspiring POV on this is due to Witten, who interprets the entire partition function as the wave function in a standard geometric quantization picture for the middle cohomology of the CY (or more suggestively, of the moduli of CYs). This is also behind the Givental quantization formalism for the higher genus A-model, where the issue is not defining the invariants
but finding a way to calculate them.

Anyway I don't know a totally satisfactory mathematical formalism for the meaning of this partition function (and have tried to get it from many people), so would love to hear any thoughts. But the strong message from physics is that we should try to interpret this entire partition function - in particular it is this function which appears in a million different guises under various dualities (eg in gauge theory, as solution to quantum integrable systems, etc etc...)

ag.algebraic geometry - Quotient of abelian variety by an abelian subvariety

Let us work over $mathbb{C}$.

The inclusion $u colon B to A$ induces a surjection $hat{u} colon A^{vee} to B^{vee}$.
By general facts on Abelian varieties, the kernels of $u$ and $hat{u}$ have the same number of connected components. Since $u$ is injective, its kernel is trivial, so it follows $ker hat{u}=(ker hat{u})_0$; in other words $ker hat{u}$ is an Abelian subvariety of $A^{vee}$.

Therefore we have an exact sequence of Abelian varieties
$$0 to ker hat{u} to A^{vee} to B^{vee} to 0.$$ By dualizing it, we obtain $$0 to B to A to (ker hat{u})^{vee} to 0,$$
that is $C = (ker hat{u})^{vee}$.

gr.group theory - Does $mathrm{Aut}(mathrm{Aut}(...mathrm{Aut}(G)...))$ stabilize?

I don't know about non-stabilizing, but rigidity provides many examples that stabilize quickly.

1) Let π be the fundamental group of a finite volume hyperbolic manifold M of dimension ≥ 3 with no symmetries (that is, no nontrivial self-isometries). Negative curvature implies that π is centerless, so the map π -> Aut(π) is injective. Mostow-Prasad rigidity says that Out(π) = Isom(M), so the lack of isometries implies that Out(π) is trivial and Aut(π) = π. [This works verbatim for lattices in higher-rank semi-simple Lie groups subject to appropriate conditions.]

2) Let π=F_d be a free group of rank 2≤d<∞. Then Aut(F_n) is a much larger group; however, Dyer-Formanek showed that Out(Aut(F_n)) is trivial. Thus since Aut(F_n) is clearly centerless, we have Aut(Aut(F_n)) = Aut(F_n).

3) Interpolating between these two examples, if π=π₁(S_g) is the fundamental group of a surface of genus g≥2, then Aut(π) is the so-called "punctured mapping class group" Mod_g,*, which is much bigger than π. Ivanov proved that Out(Mod_g,*) is trivial, and since Mod_g,* is again centerless, we have Aut(Aut(π₁(S_g))) = Aut(π₁(S_g)).

In each of these cases, rigidity in fact gives stronger statements: Let H and H' be finite index subgroups of G = Aut(F_n) or Mod_g,*. (This class of groups can be widened enormously, these are just some examples.) Then any isomorphism from H to H' comes from conjugation by an element of G, by Farb-Handel and Ivanov respectively. In particular, Aut(H) is the normalizer of H in G. Rigidity gives the same conclusion for H = π₁(M) as in the first example and G = Isom(Hⁿ) [which is roughly SO(n,1)]. It seems that by carefully controlling the normalizers, you could use this to construct examples that stabilize only after n steps, for arbitrary large n.

---

Edit: I find the examples of D₈ and D_∞ unsatisfying because even though Inn(D) is a proper subgroup of Aut(D), we still have Aut(D) isomorphic to D. Here is a general recipe for building similarly liminal examples. Let G be an infinite group with no 2-torsion so that Aut(G) = G and H¹(G;Z/2Z) = Z/2Z. (Edited: For example, by rigidity, any hyperbolic knot complement with no isometries has these properties; by Thurston, most knot complements are hyperbolic.) The condition on the 2-torsion implies that for any automorphism G x Z/2Z -> G x Z/2Z, the composition

G -> G x Z/2Z -> G x Z/2Z -> G

is an isomorphism. From this we see that Aut(G x Z/2Z) / G = H¹(G;Z/2Z) = Z/2Z. By examination the extension is trivial, and thus Aut(G x Z/2Z) = G x Z/2Z. However, the image Inn(G x Z/2Z) is the proper subgroup G.

Comments: looking back, this feels very close to your original example of R x Z/2Z. Interesting that it's (seemingly) much harder to find group-theoretic conditions to force the behavior the way you want, while topologically it's easy.

Also, if you instead take G with H¹(G;Z/2Z) having larger dimension, say H¹(G;Z/2Z) = (Z/2Z)², this blows up quickly. You get Aut(G x Z/2Z) = G x (Z/2Z)², but then Aut(Aut(G x Z/2Z)) is the semidirect product of H¹(G;Z/2Z²) = (Z/2Z)⁴ with Aut(G) x Aut(Z/2Z²) = G x GL(2,2). Already the next step seems very hard to figure out. However, if you had enough control over the finite quotients of G, perhaps you could show that the linear parts of these groups don't get "entangled" with the rest, so that the automorphism groups would act like a product of G x (Z/2Z)ⁿ with something else, with n going to infinity. If so, this could yield an example where the isomorphism types of the groups never stabilize.

ag.algebraic geometry - Homology class orthogonal to image of Chern characters?

As Tony Pantev points out, it is easy to make an example by taking non-algebraic classes.

If you impose that $t$ is algebraic, and take $X=Y$, you are very close to stating Grothendieck's Conjecture D. Let $X$ be smooth and projective. The conjecture is that any algebraic class in $H^*(X)$, which is orthogonal to all algebraic classes, is zero torsion.

kt.k theory homology - Any reason why K_23(Z) has order 65520?

More generally, if $F$ is a number field with ring of integers $mathfrak{o}$, and $zeta_F^ast(m)$ is the first nonzero coefficient in the Taylor expansion of $zeta_F$ at $m$, then Lichtenbaum (and Quillen) conjectured that $|zeta_F^ast(1-i)|=frac{# K_{2i-2}(mathfrak{o})_{text{tors}}}{# K_{2i-1}(mathfrak{o})_{text{tors}}}$, times a regulator and some power of 2 (which I believe is not understood in general, although some progress was made on this in Ion Rada's PhD thesis). Hence, odd $K$ groups are related to the denominators of the Bernoulli numbers, and the even ones are related to the numerators. Also, not much cancellation occurs; I think the two $K$-groups can only share factors of 2.

The Voevodsky-Rost theorem might prove the Lichtenbaum conjecture, but I haven't seen anyone come out and say definitely that this is the case.

I don't have much intuition for this, except that the $K$-groups seem to be objects that like to map into étale cohomology groups. In this paper (link to MathSciNet), Soulé constructs Chern class maps from certain $K$-groups to étale cohomology groups. Furthermore, these maps frequently have small (or trivial) kernels and cokernels. I suppose the idea, then, is that $K$-theory is supposed to be a slightly better behaved version of étale cohomology, at least for the purpose of understanding zeta functions.

The rank of $K$-groups of rings of integers was computed by Quillen in the early 70's: it's rank 1 in dimension 0, rank $r_1+r_2-1$ in dimension 1 (Dirichlet's unit theorem), rank 0 in even dimensions $>0$, rank $r_1+r_2$ in dimensions $1pmod 4$ except 1, and rank $r_2$ in dimensions $3pmod 4$.

at.algebraic topology - finite generated group realized as fundamental group of manifolds

Theorem. Every finitely presentable group is the fundamental group of a closed 4-manifold.

Sketch proof. Let $langle a_1,ldots,a_mmid r_1,ldots, r_nrangle$ be a presentation. By van Kampen, the connected sum of $m$ copies of $S^1times S^3$ has fundamental group isomorphic to the free group on $a_1,ldots, a_m$. Now we can quotient by each relation $r_j$ as follows. Realise $r_j$ as a simple loop. A tubular neighbourhood of this looks like $S^1times D^3$. Do surgery and replace this tubular neighbourhood with $S^2times D^2$. This kills $r_j$. QED

There are many restrictions on 3-manifold groups. One of the simplest arises from the existence of Heegaard splittings. It follows easily that if $M$ is a closed 3-manifold then $pi_1(M)$ has a balanced presentation, meaning that $nleq m$.

Other obstructions to being a 3-manifold group were discussed in this MO question.

ct.category theory - Sites which are stacks over themselves

I don't have an answer to your question, but I'm going to post whatever thoughts I had about it. Maybe something here will help someone answer the question, or at least help more people understand what's involved. I'm sorry that it's come out so long.

---

Definitions
(skip this unless you suspect we mean different things by "(pre)stack")

A functor $Fto C$ is a fibered category if for every arrow $f:Uto X$ in $C$ and every object $Y$ in $F$ lying over $X$, there is a cartesian arrow $Vto Y$ in $F$ lying over $f$ (see Definition 3.1 of Vistoli's notes). This arrow is determined up to unique isomorphism (by the cartesian property), so I'll call $V$ "the" pullback of $Y$ along $f$ and maybe denote it $f^*Y$. A fibered category is roughly a "category-valued presheaf (contravariant functor) on $C$".

Given an object $X$ in $C$, let $F(X)$ be the subcategory of objects in $F$ lying over $X$, with morphisms being those morphisms in $F$ which lie over the identity morphism of $X$. I'll call $F(X)$ the "fiber over $X$." Given a morphism $f:Uto X$ in $C$, let $F(Uto X)$ be "the category of descent data along $f$," whose objects consist of an element $Z$ of $F(U)$ and an isomorphism $sigma:p_2^*Zto p_1^*Z$ (where $p_1,p_2:Utimes_XUto U$ are the projections) satisfying the usual cocycle condition over $Utimes_XUtimes_XU$ (see Definition 4.2 of Vistoli's notes). A morphism in $F(Uto X)$ is a morphism $Zto Z'$ in $F(U)$ such that the following square commutes:
$begin{matrix}
p_2^*Z & xrightarrow{sigma} & p_1^*Z \
downarrow & & downarrow\
p_2^*Z' & xrightarrow{sigma'} & p_1^*Z'
end{matrix}$

Suppose $C$ has the structure of a site. Then we say that $F$ is a prestack (resp. stack) over $C$ if for any cover $Uto X$ in $C$, the functor $F(X)to F(Uto X)$ given by pullback is fully faithful (resp. an equivalence). Roughly, a prestack is a "separated presheaf of categories" and a stack is a "sheaf of categories" over $C$.

---

The domain fibration (not your question, but related)

Consider the domain functor $Arr(C)to C$ given by $(Xto Y)mapsto X$. You can check that a cartesian arrow over $f:Uto X$ is a commutative square
$begin{matrix}
U & xrightarrow{f} & X \
downarrow & & downarrow\
Y & = & Y
end{matrix}$
If I haven't made a mistake,

• This fibered category is a prestack iff every cover $Uto X$ is an epimorphism.


• It is a stack if furthermore every cover $Uto X$ is the coequalizer of the projection maps $p_1,p_2:Utimes_XUto U$. This last condition is equivalent to saying that every object $Y$ of $C$ satisfies the sheaf axiom with respect to the morphism $Uto X$. In particular, the domain fibration is a stack if and only if the topology is subcanonical.

---

The codomain fibration (your question)

Consider the codomain functor $Arr(C)to C$ given by $(Uto X)mapsto X$. You can check that a cartesian arrow over a morphism $f:Uto X$ is a cartesian square
$begin{matrix}
V & to & U \
downarrow & & downarrow\
Y & xrightarrow{f} & X
end{matrix}$
There is a general result that says that the fibered category of sheaves on a site is itself a stack (I usually call this result "descent for sheaves on a site"). If you're working with the canonical topology on a topos (where every sheaf is representable), it follows that the codomain fibration is a stack. If the topology is subcanonical, then objects are sheaves, so descent for sheaves tells you that the pullback functor is fully faithful (i.e. the codomain fibration is a prestack), but when you "descend" a representable sheaf, it may no longer be representable, so the codomain fibration may not be a stack. In your question you say that being a prestack is actually equivalent to the topology being subcanonical, but I can't see the other implication (prestack⇒subcanonical).

Supposing the codomain fibration is a prestack, saying that it is a stack roughly says that when you glue representable sheaves along a "cover relation," you get a representable sheaf, but with the strange condition that the "cover relation" you started with came from a relation where you could glue to get a representable sheaf. That is, given this diagram, where the squares on the left are cartesian ($Rightarrow$ is meant to be two right arrows), can you fill in the "?" so that the square on the right is cartesian?
$begin{matrix}
Z' & Rightarrow & Z & to & ?\
downarrow & & downarrow & & downarrow\
Utimes_XU & Rightarrow & U & to & X
end{matrix}$

A more natural (to me) condition is to ask that the only sheaves you can glue together from representable sheaves are already representable. That is, if $RRightarrow U$ is a "covering relation" (i.e. each of the maps $Rto U$ is a covering and $Rto Utimes U$ is an equivalence relation), then the quotient sheaf $U/R$ is representable. I would call such a site "closed under gluing."

For example, the category of schemes with the Zariski topology is closed under gluing (it's the "Zariski gluing closure" of the category affine schemes). The category of algebraic spaces with the etale topology is closed under gluing (it's the "etale gluing closure" of the category of affine schemes). In fact, I think that a standard structure theorem for smooth morphisms and a theorem of Artin (∃ fppf cover ⇒ ∃ smooth cover) imply that the category of algebraic spaces with the fppf topology is closed under gluing.

soft question - The work of Thurston

There are several sources for Thurston's hyperbolization theorem, some published, some not.

Off the top of my head:

1) M.Kapovich, Hyperbolic manifolds and discrete groups.

2) J. Hubbard's Teichmuller theory volume II (not yet published)

3) J. Morgan, H. Bass (eds). The Smith conjecture. (English)
Papers presented at the symposium held at Columbia University, New
York, 1979.
Pure and Applied Mathematics, 112.
Academic Press, Inc., Orlando, Fla., 1984. xv+243 pp.

For only the case of manifolds that fibre over S^1

1) J-P. Otal, The hyperbolization theorem for fibred 3-manifolds.

Of course there's also the new non-Thurston proofs using Ricci flow.

Oh, and regarding that anecdote about repelling people from a field -- I've only heard that comment attributed to one mathematician and it was in reference to Thurston's early work on foliations. I don't think that's a widely held belief, but I wasn't alive then so I'm just going on 2nd hand comments.

reference request - Manifolds of continuous mappings.

I'm planning a short course on few topics and applications of nonlinear functional analysis, and I'd like a reference for a quick and possibly self-contained construction of a structure of a Banach differentiable manifold for the space of continuous mappings $C^0(K,M)$, where $K$ is a compact topological space (even metric if it helps) and $M$ is a (finite dimensional) differentiable manifold.

A construction of a differentiable structure of Banach manifold for this space can be found e.g. in Lang's book Fundamentals of differential geometry (1999). The main tools are the exponential map and tubular nbds (having fixed a Riemannian structure on $M$. This is OK but I believe there should be something even more basic.

Does anybody have a reference for alternative constructions (not necessarily elementary) ?

special functions - modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder

$R_{k,m} equiv H_{k} ~ mod H_m$

for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial (orthogonal under the weight $w(x) = e^{-x^2}$) and $deg R_{k,m} leq m-1$. I haven't been able to find anything online, neither could compute it through the recurrence relation of Hermite polynomials...

Update:

The motivation for my question is as follows. The $m$-point Gauss-quadrature is obtained by placing the nodes at the roots of $H_m$ and choosing the weights accordingly such that integrating any polynomial (with respect to weight $w$) of order $leq 2m-1$ is exact. Now I want to know the error formula for polynomials of degree $k geq m$, especially $H_k$. By computing $H_k$ modulo $H_m$, the integration error is given by the integration of the remainder $R_{k,m}$.

real analysis - existence of antiderivatives of nasty but elementary functions

This shows that "elementary function" needs a good definition. We do NOT want to allow, for example $f(x) = 1$ when $x$ rational and $f(x) = -1$ when $x$ irrational. Even though $f^2 = 1$, this $f$ is not an algebraic function.

So, correctly defined, an elementary function is an analytic function on a domain in the complex plane, such that ...... [fill in the usual conditions]

Added later. My advice: For "elementary function" do not use the popularized form of the
definition as in Wikipedia. Instead, use a definition from
the actual mathematics papers. (Papers with proofs, not
just quickie approximate definitions for the masses.)

For example

"Integration in Finite Terms", Maxwell Rosenlicht,
The American Mathematical Monthly 79 (1972), 963--972.
Stable URL: http://www.jstor.org/stable/2318066

Everything is carried out in differential fields ... In particular, every function involved is infinitely differentiable ... None of those
"discontinuous elementary functions" mentioned in the question.
Not even $|x| = sqrt{x^2}$ is elementary.

===========

"Algebraic Properties of the Elementary Functions of Analysis",
Robert H. Risch,
American Journal of Mathematics 101 (1979) 743--759.
Stable URL: http://www.jstor.org/stable/2373917

He also works in differential fields. Some quotes:

The elementary functions of a complex variable $z$ are those analytic functions that are built up from the rational functions of $z$ by successively applying algebraic operations, exponentiating, and taking logarithms. As is well known, this class includes the trigonometric and basic inverse trigonometric functions.

[Part II]
Suppose $mathbb{C}(z, theta_1, dots, theta_m) = mathcal{D}_m$ is the abstract field, isomorphic to a field of meromorphic functions on some
region $R$ of the complex plane, ...

==========

books - Text for an introductory Real Analysis course.

I'm not a fan of the Pfaffenberger text. For example, look at the proof of the chain rule. The proof sticks to the "derivative as slope" idea, and so has to consider the special case where one derivative is zero. This isn't very elegant, and causes confusion in what should be a straightforward proof -- IMO when students are first being exposed to something as elementary as analysis, simplicity should be an overriding concern.

Apostol, Buck and Bartle, those are texts that I like pretty well. Or the lecture notes used at the University of Alberta for their honours calculus sequence Math 117, 118, 217, 317 (available on-line) -- pretty well based on Apostol.

There's a few subtle issues going on here. Some departments view analysis as something people learn after they go through a service-level calculus sequence. Some departments treat calculus as part of an analysis sequence -- ie students only see calculus through the eyes of analysis. What book you choose is largely determined by what path your department is comfortable with.

graph theory - How to estimate the growth of the probability that $G(n, M)$ contains a $k$-clique

You might take a look at Chapter VII of Bollobas. In particular,
Theorem VII.1.7 -- which is simple enough that he doesn't bother providing a proof -- states that the expected number of $k$-cliques in $G(n,M)$ is, setting $N={n choose 2}$ and $K={k choose 2}$,
$$
{n choose k} {n-K choose M- K} {N choose M}^{-1}.
$$
Also, Theorem VII.3.7 states that if $M=o(n^{-2/(k-1)})$ then with probability tending to one, $G_{n,M}$ contains no $k$-clique, whereas if $M/n^{-2/(k-1)} to infty$ then with probability tending to one $G_{n,M}$ does contain a $k$-clique. I know this doesn't fully answer your question but it may help.

Incidentally, (you probably already realize that) it is a priori possible (though I don't think it is the case) that, for example, $t_k(M+1)-t_k(M) geq frac{1}{mathrm{poly}(n)}$ for all ${k choose 2}leq M leq lceil frac{(k-1)N}{k}rceil$, since all we really know by Turán is that
$$
sum_{M=K}^{lceil(k-1)N/krceil} (t_k(M+1)-t_k(M)) = 1.
$$

triangulated categories - Why do people "forget" Verdier abelianization functor?(Looking for application)

I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal cohomological functor) is really powerful and useful formalism. The professor assigned many problems concerning the property of localization functor in triangulated category.He strongly suggested us using abelianization functor to do these problems

If we do these problems in triangulated category, we have to work with various axioms TRI to TRIV which are not very easy to deal with. But if we use Verdier abelianization functor, we can turn the whole story to the abelian settings. Triangulated category can be embedded to Frobenius abelian category(projectives and injectives coincide). Triangulated functors become exact functor between abelian categories. Then we can work in abelian category. Then we can easily go back(because objects in triangulated category are just projectives in Frobenius abelian category, we can use restriction functor). In this way, it is much easier to prove something than Verider did in his book.

My question is:

1. What makes me surprised is that Verdier himself even did not use Abelianization in his book to prove something. I do not know why?(Maybe I miss something)




2. I wonder whether there are any non-trivial application of Verdier abelianization functor in algebraic geometry or other fields?

Thank you

homological algebra - Sums of injective modules, products of projective modules?

1. Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?




2. Analogously, under what assumptions on R does a countable product of projective left R-modules necessarily have a finite projective dimension?

These questions arise in the study of the coderived and contraderived categories of (CDG-)modules, or, if one wishes, the homotopy categories of unbounded complexes of injective or projective modules.

There are some obvious sufficient conditions and some less-so-obvious ones. For both #1 and #2, it clearly suffices that R have a finite left homological dimension.

More interestingly, in both cases it suffices that R be left Gorenstein, i.e., such that the classes of left R-modules of finite projective dimension and left R-modules of finite injective dimension coincide.

For #1, it also suffices that R be left Noetherian. For #2, it suffices that R be right coherent and such that any flat left module has a finite projective dimension.

Any other sufficient conditions?

The category of finite locally-free commutative group schemes

Voici tout ce qu'il y a à savoir sur cette catégorie FL/S.

En toutes généralités, c'est une catégorie exacte au sens de Quillen. Plus précisément, le plongement de FL/S dans la catégorie des faisceaux de groupes abéliens fppf sur $S$ font de FL/S une sous-catégorie stable par extensions dans cette catégorie abélienne de faisceaux fppf. Dans une catégorie exacte on dispose d'une notion de monomorphismes et épimorphismes stricts: ce sont ceux qui peuvent s'insérer dans une suite exacte. Alors, si $f:Grightarrow H$ est un morphisme de schémas en groupes, $f$ est un monomorphisme strict si et seulement si c'est une immersion fermée. De plus, $f$ est un épimorphisme strict si et seulement si c'est un morphisme fidèlement plat.

En général cette catégorie exacte n'est pas abélienne. Comme rappelé précédemment c'est cepdendant le cas si $S$ est le spectre d'un corps.

Maintenant supposons que $S$ soit le spectre d'un anneau de valuation d'inégales caractéristiques que je note $mathcal{O}_K$.

Lorsque $e_{K/mathbb{Q}_p} < p-1$ Raynaud a montré que c'est une catégorie abélienne. De plus, le foncteur fibre générique $Gmapsto Gotimes K$ est pleinement fidèle et identifie FL/S à une sous-catégorie abélienne de FL/$spec(K)$ (i.e., après un choix d'une clôture algébrique $overline{K}$ de $K$, une sous-catégorie abélienne de la catégorie des $Gal(overline{K}|K)$-modules discrets finis en tant que groupe abélien).

Lorsque $e geq p-1$ le résultat précédent est faux. Néanmoins on a le résultat suivant: dans FL/S tout morphisme possède un noyau et un conoyau. Plus précisément, si $f$ est un morphisme dans FL/S alors le platifié de $ker f$ ($ker f =$ noyau usuel dans la catégorie des schémas en groupes non-nécessairement plats) (platifié= on tue la $p$-torsion) est un noyau dans $FL/S$ du morphisme $f$. On construit de même l'image de $f$ comme adhérence schématique de l'image en fibre générique. Cependant, la catégorie précédente n'est pas abélienne. Soit en effet $K=mathbb{Q}_p (zeta_p)$ et
$$
f: mathbb{Z}/pmathbb{Z} longrightarrow mu_p
$$
le morphisme qui à $bar{1}in mathbb{Z}/pmathbb{Z}$ associe $zeta_pin mu_p$. Alors, $f$ est un isomorphisme en fibres génériques. On en déduit que dans FL/S les noyaux et conoyaux de $f$ sont nuls. Ce n'est cependant pas un isomorphisme !