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genus

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, genus of a curve. There is also genus of a lattice.

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Idea

Basic

For RR a (commutative) ring, an RR-valued genus is a ring homomorphism into RR

σ:Ω R \sigma : \Omega_\bullet \to 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.

In stable homotopy theory and generalized cohomology theory

Since the cobordism ring is the ring of coefficients of the corresponding universal Thom spectrum, e.g MUM U, MSOM SO, so a genus may also be written as a ring homomorphism of the form

MSO *R M SO_\ast \longrightarrow R

or

MU *R M U_\ast \longrightarrow R

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 EE with coefficient ring RR

MSOE M SO \longrightarrow E

or

MUE. M U \longrightarrow E \,.

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

* BSO 0 σ J BGL 1(E) EMod \array{ \ast && \longleftarrow && B SO \\ & {}_{\mathllap{0}}\searrow & \swArrow_\sigma & \swarrow_{\mathrlap{J}} \\ && B GL_1(E) \\ && \downarrow \\ && E Mod }

(a trivialization of the EE-(∞,1)-module bundle associated to the spherical fibration given by the J-homomorphism) and under forming homotopy colimits in EE(∞,1)Mod this becomes an EE-linear map

MSOEE M SO \wedge E \longleftarrow E

hence a map

MSOE. M SO \longleftarrow E \,.

At least in some important cases, genera seem to be naturally understood as encoding sigma-model quantum field theories. For GG some structure, the Thom spectrum MGM 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 RR itself may then be thought of as a classifying space for a certain class of QFTs, and hence the genus σ:MGR\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 MSpinKOM 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 MStringtmfM String \to tmf, which can be thought of as sending a manifold with string structure to the corresponding (2,1)-supersymmetric EFT (“heterotic string”).

Properties

Rationalization

When the coefficient ring RR does not have additive torsion, then any ring homomorphism

ϕ:Ω U/SOR \phi \colon \Omega^{U/SO}_\bullet \longrightarrow R

is detrmined already by its rationalization

ϕ:Ω U,SOR \phi \colon \Omega^{U,SO}_\bullet\otimes \mathbb{R} \longrightarrow R \otimes \mathbb{Q}

which is traditionally denoted by the same symbol. The rational cobordism rings in turn are known to be polynomial rings

Ω U[P 1,P 2,] \Omega^U_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^1,\mathbb{C}P^2, \cdots ]
Ω SO[P 2,P 4,] \Omega^{SO}_\bullet\otimes \mathbb{Q} \simeq \mathbb{Q}[\mathbb{C}P^2,\mathbb{C}P^4, \cdots ]

whose generators are identified with the cobordism classes of the manifolds which are the complex projective spaces, as indicated.

Logarithm and Characteristic series

Definition in components

Given a (rational) genus ϕ:Ω U,SUR\phi \colon \Omega^{U,SU}_\bullet\otimes \mathbb{Q} \to R \otimes \mathbb{Q} one defines (we follow (Hopkins 94))

  1. its logarithm to be the formal power series over RR \otimes \mathbb{Q} given by

    log ϕ(z) nϕ(P n)z n+1n+1; log_\phi(z) \coloneqq \sum_{n \in \mathbb{N}} \phi(\mathbb{C}P^n) \frac{z^{n+1}}{n+1};
  2. its characteristic series (or Hirzebruch series) to be the formal power series over RR \otimes \mathbb{Q}

    K ϕ(z)zexp ϕ(z), K_\phi(z) \coloneqq \frac{z}{\exp_\phi(z)} \,,

    where exp ϕ\exp_\phi is the inverse of the logarithm;

  3. its characteristic class as the universal characteristic class which via the splitting principle is fixed by its value on the universal line bundle as

    K ϕ(c 1)H (BU(1),R) K_\phi(c_1) \in H^\bullet(B U(1),R \otimes \mathbb{Q})

    where c 1H 2(BU(1),)c_1 \in H^2(B U(1), \mathbb{Z}) denotes the universal first Chern class; hence its value on a direct sum L 1L kL_1 \oplus \cdots \oplus L_k of complex line bundles is

    iK ϕ(c 1(L i)). \prod_{i} K_\phi(c_1(L_i)) \,.

Definition via orientations in generalized cohomology

Suppose that the given genus Ω SOR\Omega_\bullet^{SO} \longrightarrow R indeed comes from an orientation in generalized cohomology (as discussed above) hence from a homomorphism of E-∞ rings

β:MSOE \beta \;\colon\; M SO \longrightarrow E

for an E-∞ ring EE with homotopy groups Rπ (E)R\simeq \pi_\bullet(E). (And suppose that EE 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 EE-cohomology

β:MUMSOβE. \beta \;\colon\; M U \longrightarrow M SO \stackrel{\beta}{\longrightarrow} E \,.

Now rationally, i.e. for EE\otimes \mathbb{Q}, there is a canonical such orientation, given by the composite

α:MUMSOH𝕊E. \alpha \;\colon\; M U \longrightarrow M SO \longrightarrow H \mathbb{Q} \simeq \mathbb{S}\otimes \mathbb{Q} \stackrel{}{\longrightarrow} E \otimes \mathbb{Q} \,.

Therefore given any orientation β\beta, then its rationalization may be compared to α\alpha. Since these rational orientations are equivalently trivializations of maps to BGL 1(E)B GL_1(E \otimes \mathbb{R}), their difference is a class β/α\beta/\alpha with coefficients in GL 1(ER)GL_1(E\otimes R), hence over any space XX the difference is a class in H 0(X,π E) ×H^0(X, \pi_\bullet E\otimes \mathbb{Q})^\times.

Specifically consider the delooping X=BU(1)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)c_1(L)). Under the canonical inclusion BU(1)BUB U(1)\to B U both the above orientations β\beta and α\alpha pull back, so that we have a difference

K ββ/α(E )[[c 1(L)]]. K_\beta \coloneqq \beta/\alpha \in (E_\bullet \otimes \mathbb{Q})[ [ c_1(L) ] ] \,.

This is the Hirzebruch series of β\beta (Ando-Hopkins-Rezk 10, def. 3.10).

If FF denotes the formal group law classified via MU MSO β E MU_\bullet \to M SO_\bullet \stackrel{\beta_\bullet}{\to} E_\bullet then

K β(z)=zexp F(z) K_\beta(z) = \frac{z}{\exp_F(z)}

The Hirzebruch formula

The central theorem of (Hirzebruch 66) expresses the genus of an arbitrary (cobordism class of a) manifold XX of dimension 2n2n via the formula

ϕ(X)= i=1 nK ϕ(x i(TX)),[X] \phi(X) = \langle \prod_{i = 1}^n K_\phi(x_i(T X)), [X] \rangle

in terms of the Hirzebruch characteristic series K ϕK_\phi discussed above, and via the splitting principle:

This means that i=1 nK ϕ(x i(TX))\prod_{i = 1}^n K_\phi(x_i(T X)) is the function of Chern classes c kc_k (i.e. Pontryagin classes P 2kP_{2k} and Euler classes χ\chi) obtained by rewriting the polynomial in the x ix_i (the “Chern roots”) as a polynomial in elementary symmetric polynomials σ k(x 1,,x n)\sigma_k(x_1,\cdots, x_n) and then substituting for each of these by c k(TX)c_k(T X).

(see also e.g. ManifoldAtlas – Genera – 4.1 Construction).

Examples

Todd genus

The Todd genus us the genus with logarithm

log Todd(1x)= nx nn -log_{Todd}(1-x) = \sum_{n \in \mathbb{N}} \frac{x^n}{n}

Signature genus

The signature genus;

A^\hat A-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 A^\hat A-genus is

K A^(e) =ze z/2e z/2 =exp( k2B kkz kk!), \begin{aligned} K_{\hat A}(e) & = \frac{z}{e^{z/2} - e^{-z/2}} \\ &= \exp\left( - \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!} \right) \end{aligned} \,,

where B kB_k is the kkth Bernoulli number (Ando-Hopkins-Rezk 10, prop. 10.2).

The A^\hat A-genus is an integer on manifolds with spin structure.

Elliptic genus

An elliptic genus is one whose logarithm is given by

log ell(z)= 0 z(12δt 2+ϵt 4) 1/2dt log_{ell}(z) = \int_0^z (1-2 \delta t^2 + \epsilon t^4)^{-1/2} d t

for constants δ,ϵ\delta, \epsilon with non-degenerate values δ 2ϵ\delta^2 \neq \epsilon and ϵ=0\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 ϵ\epsilon and δ\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 (Γ 0(2))MF_\bullet(\Gamma_0(2)). This “universal” elliptic genus us the Witten genus.

Witten genus

The Witten genus

w:Ω SO[[q]] w \colon \Omega^{SO}_\bullet \longrightarrow \mathbb{Q}[ [q] ]

is the genus with coefficients in the power series ring [[q]]\mathbb{Q}[ [ q ] ] with characteristic series given by

K w(z) =z/2sinh(z/2) n1(1q n) 2(1q ne z)(1q ne z) =exp( k2G kz kk!), \begin{aligned} K_w(z) & = \frac{z/2}{sinh(z/2)} \prod_{n \geq 1} \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})} \\ & = \exp\left( \sum_{k \geq 2} G_k \frac{z^k}{k!} \right) \end{aligned} \,,

where G kG_k are the Eisenstein series (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10, prop. 10.9). (Notice that the constant term in G kG_k is proportional to the kkth 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 [[q]]\mathbb{Z}[ [ q ] ]

On manifolds with rational string structure it takes values in (the qq-expansion of) modular forms for SL 2()SL_2(\mathbb{Z}), meaning that setting q=e 2πiτq = e^{2 \pi i \tau} then as a function ff of the parameter τ\tau taking vakues in the upper half plane the Witten genus satisfies

f(1/τ)=(τ) nf(τ). f(-1/\tau) = (-\tau)^n f(\tau) \,.

Finally on manifolds with actual string structure it takes values in topological modular forms. See at Witten genus for more.

Non-examples

Euler characteristic

The Euler characteristic Xχ(X)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 D = d + d^\dagger)

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge

References

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

  • John Arthur Todd, The arithmetical invariants of algebraic loci, Proc. London Math. Soc. (2), Ser. 43, 1937, 190–225.

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

Discussion in terms of orientations in generalized cohomology and specifically for the A-hat genus and the Witten genus is in

Revised on July 19, 2014 03:21:53 by Urs Schreiber (89.204.130.28)