this entry is about the notion of genus in algebraic topology/cohomology. For classification of surfaces see instead the (related) entry genus of a surface.
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $R$ a (commutative) ring, an $R$-valued genus is a ring homomorphism into $R$
from a cobordism ring for cobordisms with specified structure; typical choices being orientation or stable complex structure. Often the rationalization of such a morphism is meant, see below at Properties – Rationalization.
To emphasize that this is indeed a ring homomorphism and hence in particular respects the multiplicative structure, a genus is sometimes (especially in older literature) synonymously called a multiplicative genus.
Since the cobordism ring is the ring of coefficients of the corresponding universal Thom spectrum, e.g $M U$, $M SO$, so a genus may also be written as a ring homomorphism of the form
or
respectively. Written this way it is immediate that genera arise naturally as the value on homotopy groups (the “decategorification” or “de-homotopification”) of homomorphisms of E-∞ ring spectra from an actual universal Thom spectrum to some E-∞ ring $E$ with coefficient ring $R$
or
This in turn induces multiplicative morphisms of the cohomology theories represented by these spectra (the domain being hence cobordism cohomology theory), and these multiplicative maps are the “families version” of the given genus/index (Hopkins 94, section 3).
Such homomorphisms in turn arise naturally from universal orientations in generalized E-cohomology. Namely such an orientation is a homotopy of the form
(a trivialization of the $E$-(∞,1)-module bundle associated to the spherical fibration given by the J-homomorphism) and under forming homotopy colimits in $E$(∞,1)Mod this becomes an $E$-linear map
hence a map
At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For $G$ some structure, the Thom spectrum $M G$ is the classifying space of manifolds with G-structure, and hence may be thought of as classifying target spaces for sigma-models. The codomain spectrum $R$ itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus $\sigma : M G \to R$ can be thought of as assigning to any target space the corresponding sigma-model.
This is for instance the case at least over the point for the A-hat genus $M Spin \to K O$, which may be thought of as sending manifolds with spin structure to the corresponding (1,1)-supersymmetric EFT (“spinning particle”); and for the Witten genus $M String \to tmf$, which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).
When the coefficient ring $R$ does not have additive torsion, then any ring homomorphism
is detrmined already by its rationalization
which is traditionally denoted by the same symbol. The rational cobordism rings in turn are known to be polynomial rings
whose generators are identified with the cobordism classes of the manifolds which are the complex projective spaces, as indicated.
Given a (rational) genus $\phi \colon \Omega^{U,SU}_\bullet\otimes \mathbb{Q} \to R \otimes \mathbb{Q}$ one defines (we follow (Hopkins 94))
its logarithm to be the formal power series over $R \otimes \mathbb{Q}$ given by
its characteristic series (or Hirzebruch series) to be the formal power series over $R \otimes \mathbb{Q}$
where $\exp_\phi$ is the inverse of the logarithm;
its characteristic class as the universal characteristic class which via the splitting principle is fixed by its value on the universal line bundle as
where $c_1 \in H^2(B U(1), \mathbb{Z})$ denotes the universal first Chern class; hence its value on a direct sum $L_1 \oplus \cdots \oplus L_k$ of complex line bundles is
Suppose that the given genus $\Omega_\bullet^{SO} \longrightarrow R$ indeed comes from an orientation in generalized cohomology (as discussed above) hence from a homomorphism of E-∞ rings
for an E-∞ ring $E$ with homotopy groups $R\simeq \pi_\bullet(E)$. (And suppose that $E$ defines a complex oriented cohomology theory.)
This defines (Ando-Hopkins-Rezk 10, prop. 2.11) a universal orientation of real vector bundles and hence of complex vector bundles and hence of complex line bundles in $E$-cohomology
Now rationally, i.e. for $E\otimes \mathbb{Q}$, there is a canonical such orientation, given by the composite
Therefore given any orientation $\beta$, then its rationalization may be compared to $\alpha$. Since these rational orientations are equivalently trivializations of maps to $B GL_1(E \otimes \mathbb{R})$, their difference is a class $\beta/\alpha$ with coefficients in $GL_1(E\otimes R)$, hence over any space $X$ the difference is a class in $H^0(X, \pi_\bullet E\otimes \mathbb{Q})^\times$.
Specifically consider the delooping $X= B U(1)$ of the circle group. For this the cohomology ring is the power series ring in a single variable (the universal first Chern class $c_1(L)$). Under the canonical inclusion $B U(1)\to B U$ both the above orientations $\beta$ and $\alpha$ pull back, so that we have a difference
This is the Hirzebruch series of $\beta$ (Ando-Hopkins-Rezk 10, def. 3.10).
If $F$ denotes the formal group law classified via $MU_\bullet \to M SO_\bullet \stackrel{\beta_\bullet}{\to} E_\bullet$ then
The central theorem of (Hirzebruch 66) expresses the genus of an arbitrary (cobordism class of a) manifold $X$ of dimension $2n$ via the formula
in terms of the Hirzebruch characteristic series $K_\phi$ discussed above, and via the splitting principle:
This means that $\prod_{i = 1}^n K_\phi(x_i(T X))$ is the function of Chern classes $c_k$ (i.e. Pontryagin classes $P_{2k}$ and Euler classes $\chi$) obtained by rewriting the polynomial in the $x_i$ (the “Chern roots”) as a polynomial in elementary symmetric polynomials $\sigma_k(x_1,\cdots, x_n)$ and then substituting for each of these by $c_k(T X)$.
(see also e.g. ManifoldAtlas – Genera – 4.1 Construction).
The Todd genus us the genus with logarithm
The signature genus;
The A-hat genus is the index of a Dirac operator coming from a spin bundle in KO-theory. It is given by the characteristic series
The characteristic series of the $\hat A$-genus is
where $B_k$ is the $k$th Bernoulli number (Ando-Hopkins-Rezk 10, prop. 10.2).
The $\hat A$-genus is an integer on manifolds with spin structure.
An elliptic genus is one whose logarithm is given by
for constants $\delta, \epsilon$ with non-degenerate values $\delta^2 \neq \epsilon$ and $\epsilon = 0$.
For degenerate choices this reproduces the signature genus and the A-hat genus above, see at elliptic genus for more. For non-degenrate values one may regard $\epsilon$ and $\delta$ as values of modular forms of the same name and hence regard all elliptic genera together as one single genus with coefficients in $MF_\bullet(\Gamma_0(2))$. This “universal” elliptic genus us the Witten genus.
The Witten genus
is the genus with coefficients in the power series ring $\mathbb{Q}[ [ q ] ]$ with characteristic series given by
where $G_k$ are the Eisenstein series (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10, prop. 10.9). (Notice that the constant term in $G_k$ is proportional to the $k$th Bernoulli number, so that indeed the exponential expression matches that for the A-hat genus above.)
On manifolds with spin structure whe Witten genus takes values in $\mathbb{Z}[ [ q ] ]$
On manifolds with rational string structure it takes values in (the $q$-expansion of) modular forms for $SL_2(\mathbb{Z})$, meaning that setting $q = e^{2 \pi i \tau}$ then as a function $f$ of the parameter $\tau$ taking vakues in the upper half plane the Witten genus satisfies
Finally on manifolds with actual string structure it takes values in topological modular forms. See at Witten genus for more.
The Euler characteristic $X \mapsto \chi(X)$ is close to being a genus, but is not cobordism invariant
(this is the index of the Dirac operator $D = d + d^\dagger$)
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
The abstract concept of genus is due to Friedrich Hirzebruch. It had evolved out of the older concept of (arithmetic) genus of a surface via the concept of Todd genus introduced in
An review of the history is at the beginning of (Hirzebruch-Kreck 09)
The theory of multiplicative sequences and characteristic series of genera is due to
Friedrich Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete, 9, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, New York, 1966. MR0202713 (34 #2573) Zbl 0843.14009
Reviews include
Serge Ochanine, What is… an elliptic genus?, Notices of the AMS, volume 56, number 6 (2009) (pdf)
Friedrich Hirzebruch, Matthias Kreck, On the concept of genus in topology and complex analysis, Notices of the AMS, volume 56, number 6 (2009) pdf
Michael Hopkins, section 2 of Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)
Manifold Atlas, Formal group laws and genera
Wikipedia, Genus of a multiplicative series
Discussion in terms of orientations in generalized cohomology and specifically for the A-hat genus and the Witten genus is in
Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850
Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)