Local Fibration in Singularity Theory

• Local fibration is a local structure near singularities, exhibiting Milnor-type tube and sphere fibrations based on controlled topology and transversality.
• It employs conditions such as d-regularity and stratification to achieve open-book structures and fiber equivalence in varying analytic settings.
• Advanced methods extend these fibrations to settings with positive-dimensional discriminants and to broader contexts like Lie algebroid and Grothendieck-topological fibrations.

Searching arXiv for relevant papers on local fibration and related Milnor-type fibrations. Local fibration denotes a fibration structure defined in a sufficiently small neighborhood of a distinguished point, subset, or local base, rather than on an entire global space. In singularity theory, the term usually refers to Milnor-type tube or sphere fibrations for analytic or semi-algebraic maps near a critical value, either over the complement of the zero value or over the complement, or a stratification, of a positive-dimensional discriminant. In broader usage, the same expression is generalized to Grothendieck-topological fibrations, Lie algebroid fibrations, and quantitative bundle theorems under local curvature and regularity assumptions (Dutertre et al., 2014, Ribeiro et al., 2017, Bartoli et al., 18 Jul 2025, Huang et al., 23 May 2026).

1. Milnor’s local model and its real-analytic variants

The classical prototype is Milnor’s fibration for a holomorphic germ

$f:(\mathbb C^{n+1},0)\to(\mathbb C,0),$

for which, when $\varepsilon>0$ is sufficiently small, the normalized map

$$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$

is a smooth locally trivial fibration. Milnor also proved the corresponding tube fibration

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$

and Lê Dũng Tráng showed that this tube fibration is fiber-equivalent to the sphere fibration. The fiber

$F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$

is the Milnor fiber; in the isolated critical point case it has the homotopy type of a bouquet of $n$-spheres, and

$\chi(F_t)=1+(-1)^n\mu_f,$

where $\mu_f$ is the Milnor number (Dutertre et al., 2014).

For a real analytic germ

$$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$

Milnor proved an analogous tube fibration under isolated singularity assumptions: $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$ He also constructed a vector field on the complement of the zero set $\varepsilon>0$0 whose flow pushes the tube fibration onto a sphere fibration

$\varepsilon>0$1

where $\varepsilon>0$2. In the real case, however, the sphere fibration is not automatically given by the normalized map $\varepsilon>0$3, and the sphere-side construction is subtler because the projection on the punctured sphere is generally not proper when the link is nonempty (Dutertre et al., 2014).

For non-isolated real singularities, Milnor-type fibrations require extra regularity. The survey literature records Milnor-type conditions such as $\varepsilon>0$4 near $\varepsilon>0$5 and isolation of $\varepsilon>0$6 in $\varepsilon>0$7, where $\varepsilon>0$8 is the Milnor set, together with stronger hypotheses such as $\varepsilon>0$9-regularity and Thom $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$0-type conditions. Under such assumptions one recovers local fibrations on tubes, sphere fibrations, and open-book structures with singular binding (Dutertre et al., 2014).

2. Semi-algebraic open books and normalized sphere fibrations

A particularly explicit local-fibration criterion is given for a $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$1 semi-algebraic map

$$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$2

restricted to a smooth, compact, embedded semi-algebraic manifold $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$3 of dimension $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$4, without boundary. Writing

$$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$5

the object of study is the direction map

$$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$6

The paper formulates two conditions: condition (a), excluding bad accumulation of critical points near $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$7, and condition (b), requiring that the restriction $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$8 have no critical points on $$\frac{f}{|f|}:S^{2n+1}_\varepsilon\setminus f^{-1}(0)\to S^1$$9. Under condition (a), Theorem 2.1 states that

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$0

is a $$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$1 locally trivial fibration if and only if condition (b) holds (Dutertre et al., 2014).

The proof localizes near the origin in the target, uses the Curve Selection Lemma to obtain a proper surjective submersion

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$2

applies Ehresmann’s theorem there, composes with radial projection to $$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$3, and then glues this localized fibration with the proper submersion on the outer region

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$4

In this setting, the resulting local fibration is described as an open book structure induced by $$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$5, with binding

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$6

and pages given by the fibers

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$7

(Dutertre et al., 2014).

The same framework recovers the real local Milnor fibration when $$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$8 is a small sphere

$$f:B^{2n+2}_\varepsilon\cap f^{-1}(S^1_\delta)\to S^1_\delta,$$9

Then

$F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$0

is the generalized open book with singular binding on the small sphere. The paper further analyzes the vectors

$F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$1

the Milnor equality

$F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$2

and projected maps obtained by forgetting coordinates, thereby making the transversality mechanism concrete (Dutertre et al., 2014).

3. Positive-dimensional discriminants and singular tube fibrations

When the discriminant is strictly positive-dimensional, the classical isolated-value formulation is no longer adequate. One approach begins with the notion of a nice map germ $F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$3, meaning that both $F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$4 and $F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$5 are well-defined as set germs at $F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$6. In this setting, the discriminant is defined by

$F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$7

where $F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$8 is the boundary germ of the image. For a non-constant nice analytic germ, the Milnor-Hamm fibration is the local fibration

$F_t=f^{-1}(t)\cap B^{2n+2}_\varepsilon$9

over each connected component of the complement of the discriminant, independent of the choices of $n$0 up to diffeomorphism. A sufficient condition is

$n$1

where $n$2 is the Milnor set and $n$3. The paper also introduces $n$4-Thom regularity at $n$5 as a weaker alternative to full Thom regularity and proves that it suffices for the existence of the Milnor-Hamm fibration (Ribeiro et al., 2017).

The same work defines a stronger singular Milnor tube fibration, designed to include the stratified discriminant itself. Given a regular stratification $n$6 for a non-constant $n$7-nice analytic germ, the map

$n$8

is required to be a stratified locally trivial fibration, independent up to stratified homeomorphism of $n$9. The corresponding criterion is the stratified Milnor-set condition

$\chi(F_t)=1+(-1)^n\mu_f,$0

This formulation is explicitly broader than the Milnor-Hamm fibration, since it incorporates strata contained in the discriminant and can detect nontrivial topology even when the interesting fibers lie over discriminant strata (Ribeiro et al., 2017).

A related but differently organized theory is developed for analytic germs

$\chi(F_t)=1+(-1)^n\mu_f,$1

under the tameness condition. Here the Milnor set is defined stratwise, and tameness means, in the authors’ formulation, that the bad points of the Milnor set away from the central fiber are isolated at the origin. The main theorem states that if

$\chi(F_t)=1+(-1)^n\mu_f,$2

is a non-constant analytic germ and is tame, then $\chi(F_t)=1+(-1)^n\mu_f,$3 has a singular Milnor tube fibration

$\chi(F_t)=1+(-1)^n\mu_f,$4

and this fibration is independent, up to stratified homeomorphism, of the choices of small radii. In the isolated singular value case, tameness becomes $\chi(F_t)=1+(-1)^n\mu_f,$5-regularity, and the image is locally open: $\chi(F_t)=1+(-1)^n\mu_f,$6 The same framework also yields a composition theorem: if $\chi(F_t)=1+(-1)^n\mu_f,$7 is tame and has isolated singular value and $\chi(F_t)=1+(-1)^n\mu_f,$8 has an isolated singular point, then $\chi(F_t)=1+(-1)^n\mu_f,$9 is tame, locally open, and admits a local tube fibration (Chen et al., 2022).

A common misconception is that positive-dimensional discriminant necessarily precludes a controlled local fibration. The cited results show the opposite: the obstruction is not the positive dimension of the discriminant itself, but the failure of image/discriminant stability and the failure of transversality encoded by the Milnor set (Ribeiro et al., 2017, Chen et al., 2022).

4. Tube fibrations, sphere fibrations, and normalization by target change

For real analytic maps, two local fibrations coexist: the Milnor tube fibration

$\mu_f$0

and the sphere fibration

$\mu_f$1

where $\mu_f$2 is the discriminant and $\mu_f$3 its radial projection. The bridge between them is $\mu_f$4-regularity. Using the canonical pencil

$\mu_f$5

$\mu_f$6 is $\mu_f$7-regular when sufficiently small spheres meet each $\mu_f$8 transversely, with $\mu_f$9. The paper proves that $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$0-regularity is equivalent to the spherefication map

$$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$1

being a submersion, and hence to $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$2 being a submersion on the source sphere (Cisneros-Molina et al., 2021).

The equivalence of the two local fibrations is established by constructing a vector field $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$3 that is transverse to small spheres, transverse to Milnor tubes, and satisfies that $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$4 is constant on its integral curves. Its flow inflates the tube to the sphere while preserving the angular data, yielding equivalence of the two fiber bundles. In the linear-discriminant setting, the theorem states that if $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$5 is locally surjective, has linear discriminant, satisfies the transversality property, and is $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$6-regular in the ball $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$7, then the tube bundle and the sphere bundle are equivalent (Cisneros-Molina et al., 2021).

A later refinement shows that the failure of the naive normalized map $$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$8 is not intrinsic. For a locally surjective real analytic map-germ

$$F:(\mathbb R^n,0)\to(\mathbb R^p,0),\qquad n>p\ge 2,$$9

with isolated critical value and the transversality property, there exists a homeomorphism

$$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$0

such that

$$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$1

is a smooth locally trivial fibration, equivalent to both the Milnor-Lê fibration on the tube and the Milnor fibration on the sphere. The construction uses conic homeomorphisms $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$2 produced by vector fields

$$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$3

and the notion of $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$4-regularity, where the relevant transversality is tested against preimages of the curved rays $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$5 rather than ordinary radial lines (Molina et al., 7 Jan 2026).

This body of work isolates two separate issues. The first is existence of a tube fibration, governed by transversality to spheres and properness. The second is representability of the sphere fibration by a normalized map, governed by $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$6-regularity or, after target change, by $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$7-regularity (Cisneros-Molina et al., 2021, Molina et al., 7 Jan 2026).

5. Homotopy, coordinate projections, and Euler-characteristic formulas

Once a local fibration exists, one central problem is the extent to which its fiber is stable under natural operations. For semi-algebraic open books, composition with canonical coordinate projections preserves the fibration data. If

$$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$8

then $$F:B^n_\varepsilon\cap F^{-1}(S^{p-1}_\delta)\to S^{p-1}_\delta.$$9 inherits the good conditions from $\varepsilon>0$00, and Theorem 3.6 states that the fiber $\varepsilon>0$01 is homotopy equivalent to the original fiber $\varepsilon>0$02: $\varepsilon>0$03 For $\varepsilon>0$04, the two fibers determined by the sign choices of $\varepsilon>0$05 are likewise homotopy equivalent to the fiber of $\varepsilon>0$06. More generally, submaps obtained by selecting coordinates retain the fiber homotopy type as long as the target dimension is at least $\varepsilon>0$07 (Dutertre et al., 2014).

The same paper derives explicit Euler-characteristic relations between the fiber and zero sets of coordinate submaps. One fundamental identity is

$\varepsilon>0$08

Its final summary theorem expresses $\varepsilon>0$09 in terms of $\varepsilon>0$10, the parity of $\varepsilon>0$11, and the cardinality $\varepsilon>0$12. These formulas show that the topology of the relative link constrains the fiber, and conversely (Dutertre et al., 2014).

Survey treatments place these relations in a broader Poincaré–Hopf framework. In the complex isolated singularity case,

$\varepsilon>0$13

For real analytic mappings with isolated singularity,

$\varepsilon>0$14

whereas

$\varepsilon>0$15

For non-isolated singularities satisfying Milnor-type conditions, the survey records link identities such as

$\varepsilon>0$16

together with Szafraniec-type perturbation formulas involving the degree of a normalized gradient vector field (Dutertre et al., 2014).

Stability also appears at the level of map composition. If

$\varepsilon>0$17

with $\varepsilon>0$18, where $\varepsilon>0$19 is tame and has isolated singular value and $\varepsilon>0$20 has an isolated singular point at the origin, then $\varepsilon>0$21 is tame, locally open, and admits a local tube fibration. In particular, for a $\varepsilon>0$22-regular map

$\varepsilon>0$23

the submap

$\varepsilon>0$24

has a tube fibration, and its Milnor fiber is homeomorphic to the Milnor fiber of $\varepsilon>0$25 times an open interval (Chen et al., 2022).

6. Broader meanings of local fibration

Outside singularity theory, local fibration has several distinct technical meanings. In Lie algebroid theory, a Lie algebroid fibration is defined as a surjective Lie algebroid morphism

$\varepsilon>0$26

covering a submersion $\varepsilon>0$27 and admitting a complete Ehresmann connection. Such fibrations are not expected to be locally trivial products in the ordinary sense; the paper emphasizes instead orbit-wise local triviality, and proves that when the base is $\varepsilon>0$28, every point $\varepsilon>0$29 has a neighborhood $\varepsilon>0$30 such that

$\varepsilon>0$31

where $\varepsilon>0$32 (Brahic et al., 2010).

In categorical topos theory, local fibrations generalize ordinary fibrations by incorporating Grothendieck topologies on source and base categories. For a comorphism of sites

$\varepsilon>0$33

the local lifting property is weakened from the existence of a single cartesian lift to the existence, for every arrow $\varepsilon>0$34, of a $\varepsilon>0$35-covering family

$\varepsilon>0$36

and locally cartesian arrows

$\varepsilon>0$37

such that

$\varepsilon>0$38

This notion is strong enough to recover site-level presentations of relative toposes: a morphism of sites induces a morphism of relative toposes if and only if it is a morphism of local fibrations (Bartoli et al., 18 Jul 2025).

In Riemannian and synthetic geometry, the phrase appears in bundle theorems derived from local regularity. A noncollapsed

$\varepsilon>0$39

space with bounded diameter and satisfying the $\varepsilon>0$40-local rewinding Reifenberg condition is homeomorphic to a fiber bundle over the torus $\varepsilon>0$41. Here the local rewinding Reifenberg condition requires that the universal cover of each $\varepsilon>0$42-ball be $\varepsilon>0$43-close to Euclidean balls at all smaller scales, and it is precisely this local hypothesis that upgrades an almost submetry to a topological fiber bundle (Huang et al., 23 May 2026).

A further usage occurs in the isoresidual fibration of genus-zero meromorphic differentials. For the stratum

$\varepsilon>0$44

the residue map

$\varepsilon>0$45

is an unramified cover over the complement of the resonance arrangement

$\varepsilon>0$46

with degree

$\varepsilon>0$47

Here the “local fibration” behavior is discrete rather than Milnor-type: away from resonance hyperplanes the map is locally a finite étale cover, while over the arrangement branching and degeneration occur (Gendron et al., 2020).

Taken together, these usages show that local fibration is not a single invariant construction but a family of closely related ideas: local triviality near singularities, local lifting up to cover, local transport determined by a complete connection, and local bundle structure extracted from quantitative regularity. The common core is the replacement of global product structure by a controlled local lifting or local trivialization principle.