Simplicial First Chern Class

• Simplicial First Chern Class is a universal representative of c₁ constructed on combinatorial and simplicial models rather than on smooth manifolds.
• It is computed using methods like the Bott–Shulman–Stasheff de Rham complex, Čech nerves, and simplicial presheaf approaches to connect classical and discrete geometry.
• The approach provides explicit cocycle formulas for triangulated S¹-bundles and aids in computational topology via transgression and combinatorial techniques.

Searching arXiv for the supplied papers and closely related work on simplicial first Chern classes, simplicial Chern–Weil theory, and combinatorial/simplicial representatives. arXiv search query: "simplicial first Chern class Bott-Shulman-Stasheff Chern-Weil triangulated circle bundle coherent analytic sheaves" The simplicial first Chern class is a representative of the first Chern class $c_1$ constructed on a simplicial or combinatorial model rather than directly on a smooth manifold. In the literature, this phrase covers several closely related settings: the Bott–Shulman–Stasheff simplicial de Rham complex of the nerve $NG$ of a Lie group, Čech nerves of open covers, triangulated $S^1$-bundles and their combinatorial local formulas, and simplicial presheaves or stacks of bundles and coherent sheaves. In each case, the aim is the same: to realize the universal or geometric class $c_1$ by an explicit cocycle compatible with face maps, classifying maps, and transgression constructions (Suzuki, 2017).

1. Simplicial models for bundles and classifying spaces

For a Lie group $G$, the standard simplicial models are the nerve

$NG(q)=G^q$

and the simplicial manifold

$PG(q)=G^{q+1},$

with face operators determined by multiplication in $G$ and by omission of factors. The simplicial map

$$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$

models the universal bundle $EG\to BG$ after geometric realization. Bott–Shulman–Stasheff attach to any simplicial manifold $NG$0 the double complex

$NG$1

with

$NG$2

and total differential $NG$3. For $NG$4, one has

$NG$5

while for $NG$6,

$NG$7

Dupont’s simplicial forms $NG$8 and the integration map

$NG$9

give an equivalent cohomological model, and this is the basic framework in which simplicial characteristic classes are computed (Suzuki, 2017).

A broader geometric-categorical version replaces simplicial manifolds by smooth simplicial sets. For a Fréchet Lie group $S^1$0 and a suitable Grothendieck universe $S^1$1, one obtains a smooth Kan complex $S^1$2 together with a universal simplicial $S^1$3-bundle $S^1$4. When $S^1$5 has the homotopy type of a CW complex, there is a homotopy equivalence

$S^1$6

and a universal Chern–Weil homomorphism

$S^1$7

compatible with pullback to ordinary principal bundles (Savelyev, 2021).

2. Universal simplicial Chern–Weil construction and the degree-two class

On the universal simplicial bundle $S^1$8, Dupont’s canonical simplicial connection is

$S^1$9

where $c_1$0 are barycentric coordinates on $c_1$1 and each $c_1$2 is the pullback of the Maurer–Cartan form from the $c_1$3-th factor of $c_1$4. Its curvature is

$c_1$5

For an invariant polynomial $c_1$6, the universal simplicial Chern–Weil map is

$c_1$7

The resulting cocycles in $c_1$8 are the Bott–Shulman–Stasheff forms (Suzuki, 2017).

For $c_1$9, the first Chern class and first Chern character coincide: $G$0 and the corresponding invariant polynomial is

$G$1

In the universal simplicial context, the degree-two representative can be read off from the general formula for powers of the first Chern class. For $G$2, Suzuki writes

$G$3

a total degree-$G$4 element representing the universal first Chern class, up to the choice of normalizing constants and sign conventions. Closedness is with respect to the total differential $G$5 (Suzuki, 2017).

In the simplicial de Rham complex of $G$6, the full Chern character is realized by explicit cocycles obtained from Dupont’s universal curvature. For $G$7, the invariant polynomials

$G$8

produce the universal classes $G$9. The case $NG(q)=G^q$0 is precisely the simplicial realization of $NG(q)=G^q$1 (Suzuki, 2013).

The smooth simplicial set construction gives the same degree-two class in a different model. For $NG(q)=G^q$2 or $NG(q)=G^q$3, the universal class attached to the linear invariant polynomial $NG(q)=G^q$4 is the universal real first Chern class in

$NG(q)=G^q$5

and under $NG(q)=G^q$6 it becomes the ordinary universal $NG(q)=G^q$7 in $NG(q)=G^q$8 (Savelyev, 2021).

3. Transgression and simplicial Chern–Simons forms

Because $NG(q)=G^q$9 for $PG(q)=G^{q+1},$0, every Bott–Shulman–Stasheff cocycle on $PG(q)=G^{q+1},$1 pulls back to an exact cocycle on $PG(q)=G^{q+1},$2. If

$PG(q)=G^{q+1},$3

then

$PG(q)=G^{q+1},$4

for some $PG(q)=G^{q+1},$5-cochain $PG(q)=G^{q+1},$6. These are the simplicial Chern–Simons forms. For the first Chern class, the transgression identity is

$PG(q)=G^{q+1},$7

the direct simplicial analogue of the ordinary Chern–Simons transgression formula (Suzuki, 2017).

The torus case $PG(q)=G^{q+1},$8 is especially explicit. Writing $PG(q)=G^{q+1},$9 on $G$0, Suzuki gives the $G$1-th Chern character cocycle on $G$2 by

$G$3

For $G$4,

$G$5

The associated Chern–Simons form on $G$6 is

$G$7

and it satisfies

$G$8

For $G$9, the Chern class and Chern character formulas coincide, so this is the simplicial first Chern class together with its universal primitive (Suzuki, 2017).

A related transgressive description appears in the local truncated simplicial de Rham complex. Suzuki proves Brylinski’s conjecture by constructing a local cocycle $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$0 whose boundary maps to the Bott–Shulman–Stasheff form representing $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$1. The same mechanism applies in degree $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$2, so the universal degree-one local cochain transgresses the simplicial representative of $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$3 (Suzuki, 2013).

4. Combinatorial formulas on triangulated $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$4-bundles

A different, explicitly combinatorial meaning of simplicial first Chern class arises for triangulated circle bundles. One paper states that a “trivial calculation” based on Kontsevich’s differential $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$5-form yields “very simple rational combinatorial characteristics (we call it ‘curvature’) of a triangulated $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$6 bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class”. The curvature is “expressed in terms of cyclic word in 3-character alphabet associated to the bundle”, and “from the point of view of simplicial combinatorics the word is a canonical shelling of the total complex” (Mnev, 2011).

A more systematic formulation replaces cyclic words by necklaces. For an elementary simplicial circle bundle over an ordered simplex $$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$7, the oriented fiber circle determines a surjective word

$$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$8

well-defined up to cyclic shift, hence an oriented necklace. Proper subwords select exactly one occurrence of each symbol, and their permutation parity defines the rational parity

$$\gamma:PG\to NG,\qquad \gamma(g_0,\dots,g_q)=(g_0g_1^{-1},\,g_1g_2^{-1},\,\dots,\,g_{q-1}g_q^{-1})$$9

For an elementary simplicial circle bundle $EG\to BG$0, the local coefficient of $EG\to BG$1 is

$EG\to BG$2

where $EG\to BG$3 is the associated necklace. This defines a rational simplicial local formula for the $EG\to BG$4-th power of the first Chern class. In particular, for $EG\to BG$5,

$EG\to BG$6

so the first Chern class of a triangulated circle bundle is represented on each $EG\to BG$7-simplex by a rational number computed from the parity of the associated necklace (Mnev et al., 2016).

The derivation is again Chern–Weil in origin. The necklace determines a normalized matrix $EG\to BG$8, this matrix defines a map of simplices, and pullback of Kontsevich’s cyclic invariant connection form on the universal cyclic bundle yields the local coefficient after integration. In this sense, the simplicial first Chern class is simultaneously combinatorial and universal (Mnev et al., 2016).

5. Čech nerves, coherent sheaves, and holomorphic refinements

On a complex manifold $EG\to BG$9 with a locally finite Stein cover $NG$00, the Čech nerve

$NG$01

is a simplicial complex manifold. A global holomorphic vector bundle pulls back to a simplicial bundle on this nerve, and Green’s barycentric connection

$NG$02

is an admissible simplicial connection. Its curvature $NG$03 is an endomorphism-valued simplicial $NG$04-form, and the simplicial first Chern form is

$NG$05

Dupont fiber integration sends this to the Čech–de Rham bicomplex. For a global vector bundle $NG$06, the degree-one case recovers the classical Čech representative: $NG$07 and for a line bundle $NG$08 with transition functions $NG$09,

$NG$10

Thus the simplicial first Chern class agrees with the usual first Chern class from the exponential sequence (Hosgood, 2020).

The formal foundations for this construction are developed using admissible simplicial connections and Green complexes on Čech nerves. A simplicial connection generated in degree zero is built from local connections by barycentric interpolation, and on Green complexes such connections are admissible. This is the structural condition that makes evaluation of invariant polynomials on simplicial curvature possible (Hosgood, 2020).

There is also a purely holomorphic version in which the Chern character is a map of simplicial presheaves

$NG$11

After evaluation on the Čech nerve of a good cover and passage to totalization, a bundle with local holomorphic connections and transition functions $NG$12 maps to a Čech cocycle with components

$NG$13

For a line bundle, the degree-two component is the Čech $NG$14-cocycle

$NG$15

which represents the first Chern class in Hodge cohomology (Glass et al., 2019).

This simplicial-presheaf viewpoint extends further. The simplicial presheaf $NG$16 of infinity vector bundles carries a Chern character map $NG$17, and on connected components after Čech sheafification it recovers the Toledo–Tong and O’Brian–Toledo–Tong Chern character of coherent sheaves. The degree-two piece is therefore the first Chern class in the same generalized sense, now available for stacks and equivariant settings as well (Glass et al., 2022).

6. Related meanings, variants, and modern computational realizations

The adjective “simplicial” is used in several non-equivalent ways, and the distinction matters. In toric geometry, a complete simplicial toric variety is defined by a simplicial fan, and the codimension-one part of its Chern–Schwartz–MacPherson class is

$NG$18

When $NG$19 is smooth, this equals the ordinary first Chern class $NG$20; in the simplicial, possibly singular case, it is the natural degree-one extension appearing in the CSM framework (Helmer, 2015).

In Hodge-theoretic moduli problems, one encounters a stacky or simplicial interpretation rather than a simplicial manifold of group type. For moduli of polarized Calabi–Yau manifolds, the first Chern form of the tangent bundle with the Hodge metric or the Weil–Petersson metric extends as a current to a normal-crossing compactification and represents the first Chern class of the canonical extension of the tangent bundle. The paper also explains that, from a simplicial or stacky viewpoint, these curvature forms represent the first Chern class in stack cohomology once an atlas or simplicial resolution is chosen (Liu et al., 2014).

Differential cohomology gives yet another refinement. There is a unique natural transformation

$NG$21

compatible with the underlying topological $NG$22 and with the curvature map. Although not formulated in simplicial language, this construction is model-independent and adapts naturally to simplicial settings, because it is defined by universal homotopy-theoretic properties rather than by a particular differential form model (0907.2504).

A recent computational realization makes the simplicial first Chern class entirely discrete. For a complex line bundle of local polarization vectors over a simplicial mesh $NG$23 approximating a sphere around a Weyl point, one first passes to a real rank-$NG$24 bundle, computes edge rotations $NG$25 by an orthogonal Procrustes problem, extracts angles $NG$26, and then defines on each triangle

$NG$27

This integer $NG$28-cochain is the simplicial Euler class of the real bundle and hence the simplicial first Chern class of the original complex line bundle. The resulting Chern number is gauge invariant, derivative free, structure preserving, and robust to noise, and it is used to count topological interface modes via the spectral flow–monopole correspondence (Bohlsen et al., 1 Aug 2025).

Taken together, these constructions show that the simplicial first Chern class is not a single formula but a family of equivalent realizations of $NG$29 on simplicial objects: a universal Bott–Shulman–Stasheff cocycle on $NG$30, a transgressed class on $NG$31, a local rational cocycle on triangulated $NG$32-bundles, a Čech–simplicial representative on nerves of covers, a characteristic class of simplicial presheaves and stacks, and, in recent computational work, an integer $NG$33-cochain on a simplicial mesh. The unifying principle is that $NG$34 is represented by data compatible with faces, degeneracies, and classifying maps, so that simplicial or combinatorial models recover the same characteristic class as ordinary Chern–Weil theory.