ag.algebraic-geometryat.algebraic-topologycomplex-geometrydg.differential-geometryindex-theory
Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces?
The question came up after a friend and I realized that we don't understand why one would expect to have such a formula for the signature in terms of Pontrjagin classes (i.e. the signature theorem). Any insights about this will be appreciated.
There can be many reasons for subdividing simplices, barycentrically or otherwise.
For a simplicial complex (triangulated space) there are the simplicial homology groups. These are known to be isomorphic to the singular homology groups, therefore (1) invariant under homeomorphism, and in particular (2) invariant under (not necessarily barycentric) subdivision. Before the invention of singular homology, I believe that (1) was unknown. Fact (2) was a key part of the theory. Subdivision is important simply because even if your space is made out of simplices you will sometimes care about subsets which are only unions of simplices after you cut the space up finer. In simplicial homology, excision is an easy algebraic fact, stemming from the fact that when a complex is a union of two subcomplexes then every simplex is in one or the other (or both).
In singular theory, as you know, invariance under homeomorphism is a triviality but excision requires some work. The point is that when a space is a union of two open sets then (bad news) not every singular simplex is in one or the other but (good news) simplices can be systematically replaced by combinations of smaller simplices to show that this does not matter. This is where subdivision is used, and there is no reason it has to be barycentric. It's like with the fundamental group: you might explore a space by using maps of a standard unit interval into it, but in proving the Seifert-Van-Kampen Theorem you might want to subdivide that interval into little pieces.
Barycentric subdivision also rises in PL (piecewise linear) topology in one other specific technical way that has nothing much to do with homology: regular neighborhoods. In a finite simplicial complex $K$, the smallest neighborhood of a given subcomplex $L$ that is itself a subcomplex does not in general have $L$ as a deformation retract, but this becomes true if you first barycentrically subdivide twice.
And in the interplay between categories and simplicial constructions barycentric subdivision turns up in various ways.
ADDED in response to Hatcher's answer and its comment thread:
Yes, there is a way of extending to all $n$ the pattern that begins: cut a segment in half, cut a triangle into four equal pieces using midpoints of edges ... It is sometimes called "edgewise subdivision", I believe. It may be realized for simplicial sets as follows: A simplicial set is a functor $\Delta^{op}\to Set$ where $\Delta$ is the category of standard nonempty ordered finite sets; its subdivision is obtained by composing with (the opposite of) the functor $\Delta\to\Delta$ which takes an ordered set to two copies of that set laid end to end. This leaves the realization unchanged. Applied to a standard $n$-simplex, it gives a certain subdivision with $2^n$ pieces. If $n>2$ then the pieces are not all the same shape. If $n=3$ you get a tetrahedron cut into four scaled-down models of itself sitting in the corners and four more whose union is an octahedron; these four all share an edge, the only internal edge that there is. It's not immediately clear to me what diameter estimate is available for the pieces.
This can be generalized so that you now cut an edge into $k$ equal pieces and a triangle into $k^2$ congruent pieces (almost half of which are upside down) and in general cut an $n$-simplex into $k^n$ pieces. This $k$-fold edgewise subdivision plays a role in the area of cyclic homology and related things: when a simplicial set $X$ has the kind of extra structure that makes it a cyclic set (a suitable action of a cyclic group of order $m$ on the set $X_{m-1}$ for all $m>0$) then its realization has an action of the circle group, and to make the action of the subgroup of order $k$ appear as a simplicial action you can do the $k$-fold edgewise subdivision described above.
There is also another edgewise subdivision. In this one the $1$-simplex is cut in half as before and the $2$-simplex is cut into four pieces in the following way: join the middle vertex to the midpoint of the opposite side, and join that midpoint to the midpoints of both of the other sides. This construction corresponds to the functor $\Delta\to\Delta$ that takes an ordered set to two copies of the same laid end to end but with the order reversed in one copy.
See also my answer to the recent MO question "Endofunctors of the Simplex Category".
The second edgewise subdivision that I described can be used to analyze the relationship between two definitions of algebraic $K$-theory: Quillen's $Q$-construction is essentially a subdivision of Waldhausen's $S$-construction.
First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either
a) The number $\chi (X, \mathcal O_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p_a(X)=(-1)^n(\chi (X, \mathcal O_X)-1)$, the Severi arithmetic genus, which has historical precedence but of course was defined non-cohomologically.
For example, for projective space we have $\chi (\mathbb P^n, \mathcal O_{\mathbb P^n})=1$ but $p_a (\mathbb P^n)=0$.
Hirzebruch introduced his definition mainly because it has the powerful multiplicativity property $$\chi (X\times Y,\mathcal O_{X\times Y})= \chi (X,\mathcal O_{X})\cdot \chi ( Y,\mathcal O_{ Y})$$ which certainly is a step toward the geometric interpretation you are seeking.
Another step in the right direction is that for a finite covering $X\to Y$ of degree $d$ we have the pleasant relation $\chi (X,\mathcal O_{X})=d\cdot \chi ( Y,\mathcal O_{ Y})$.
But the most important geometric property is that $\chi (X,\mathcal O_{X})$ is a birational invariant, because each number $dim_\mathbb C H^i(X,\mathcal O_{X})$ is already a birational invariant.
Arithmetic genus is reasonably easy to compute: for a hypersurface $H\subset \mathbb P^n$ of degree $d$ you have $p_a(H)=\binom {d-1}{n}$, which for $n=2$ gives the well-known elementary formula $p_a(C)=\frac {(d-1)(d-2)}{2}$ for the plane curve $C$.
[This formula (and others) can be found in Hartshorne, Chapter I, Exercise 7.2, page 54]
For a surface you have Max Noether's formula $\chi (S, \mathcal O_S)=\frac {c_1^2(S)+c_2(S)}{12}$, where $c_2(S)$ (=second Chern class of $S$) is also the purely topological Euler-Poincaré characteristic of $S$, equal to the alternating sum of the Betti numbers of the underlying toplogical space.$S_{top}$.
Finally, Fulton has given an axiomatic characterization of the arithmetic genus in algebraic geometry over an arbitrary algebraically closed field here.
In a sense it may be considered an explanation of the geometric significance of the arithmetic genus: if you want it to satisfy certain geometric properties, the definition is forced upon you.
Edit (added by Charles with Georges's permission): Fulton's axiomatic characterization may be described as follows: There is a unique assignment of a number $\mathcal{A}(X)$ to every [smooth irreducible projective variety over a fixed algebraically closed field] (hereafter simply "variety"), such that the following three axioms are satisfied:
Let $X$, $Y$, and $Z$ be (smooth) varieties of the same dimension. Suppose that $X$, $Y$, and $Z$ can be embedded as codimension-one subvarieties of a common (smooth) variety $W$, such that
Then $$\mathcal{A}(X) = \mathcal{A}(Y) + \mathcal{A}(Z) - \sum_i \mathcal{A}(V_i).$$
This assignment takes $X$ to its "Hirzebruch arithmetic number" $\mathcal{A}(X) = \chi(X, \mathcal{O}_X)$.
Best Answer
Let $S$ be a smooth algebraic complex surface. Then, there is the following relation: $$p_1=c_1(S)^2-2c_2(S)=K_S^2-2\chi_{top}(S)=3L$$ where $p_1$ is the first Pontryagin class and $L$ the L-genus.
On the other hand, cobordism theory says $p_1[S]=3\tau$ where $\tau$ is the signature of $S$.
Now (by Hodge theory)
$\tau=4\chi(\mathcal{O_S})-\chi_{top}(S)$ therefore, (pairing off with the fundamental class) the relation among $L$ and $Td$ looks like
$$K^2+\chi_{top}(S)=3\tau+3\chi_{top}(S)= 12Td(S)$$
where the second Todd class satisfies $Td(S)=1/12 (K^2+\chi_{top}(S))$