Equivariant Homotopic Distance

• Equivariant Homotopic Distance is a measure that determines the fewest G-invariant open sets required for two G-maps to be locally G-homotopic.
• It generalizes classical homotopic distance and recovers invariants like equivariant Lusternik–Schnirelmann category and equivariant topological complexity.
• The theory connects open-cover approaches with lifting-category and Whitehead–Ganea formulations, offering cohomological and dimensional bounds in equivariant settings.

Equivariant homotopic distance is an equivariant homotopy invariant that measures how many $G$-invariant local pieces are required in order for two $G$-maps to become $G$-homotopic on each piece. In the literature it appears both as $D_G(f,g)$ and as $\mathrm{HD}_G(f,g)$. It extends the homotopic distance between ordinary maps, and it recovers equivariant Lusternik–Schnirelmann category and equivariant topological complexity as special cases. In the axiomatic formulation, it is the lifting category of the pair $((f,g),\Delta)$ in the equivariant category $T_G$ of $G$-spaces and $G$-maps; in the direct formulation, it is the least size of a $G$-invariant open cover on which the two maps are locally $G$0-homotopic (Doeraene, 10 Mar 2025, Macías-Virgós et al., 2018, Daundkar et al., 28 Aug 2025).

1. Definition and ambient setting

Let $G$1 be a compact Lie group. A $G$2-space is a compactly generated Hausdorff space $G$3 equipped with a continuous left action $G$4, $G$5. A $G$6-map $G$7 between $G$8-spaces is a continuous map satisfying $G$9 for all $G$0 and $G$1. A $G$2-homotopy between $G$3-maps $G$4 is a $G$5-map $G$6 with $G$7 and $G$8, and one writes $G$9 (Daundkar et al., 28 Aug 2025).

For $D_G(f,g)$0-maps $D_G(f,g)$1, the equivariant homotopic distance $D_G(f,g)$2 is the smallest $D_G(f,g)$3 such that there exists a cover

$D_G(f,g)$4

by $D_G(f,g)$5-invariant open sets with $D_G(f,g)$6 for all $D_G(f,g)$7. If no such $D_G(f,g)$8 exists, one sets $D_G(f,g)$9. In the axiomatic notation, for $\mathrm{HD}_G(f,g)$0-maps $\mathrm{HD}_G(f,g)$1 one also writes

$\mathrm{HD}_G(f,g)$2

where $\mathrm{HD}_G(f,g)$3 is the whisker map and $\mathrm{HD}_G(f,g)$4 is the equivariant diagonal (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

The non-equivariant antecedent is the homotopic distance $\mathrm{HD}_G(f,g)$5 between continuous maps $\mathrm{HD}_G(f,g)$6, defined as the least integer $\mathrm{HD}_G(f,g)$7 such that there exists an open covering $\mathrm{HD}_G(f,g)$8 of $\mathrm{HD}_G(f,g)$9 with $((f,g),\Delta)$0 for all $((f,g),\Delta)$1; if there is no such covering, $((f,g),\Delta)$2. The equivariant definition mirrors this construction by requiring $((f,g),\Delta)$3-invariance of the cover and $((f,g),\Delta)$4-equivariance of the local homotopies (Macías-Virgós et al., 2018).

2. Sectional-category, lifting-category, and Whitehead–Ganea formulations

Let $((f,g),\Delta)$5 be the free path space on $((f,g),\Delta)$6 with the induced diagonal $((f,g),\Delta)$7-action, and let

$((f,g),\Delta)$8

For $((f,g),\Delta)$9-maps $T_G$0, form the pullback of $T_G$1 along $T_G$2:

$T_G$3

Then

$T_G$4

where $T_G$5 denotes the equivariant sectional category. A local $T_G$6-section of $T_G$7 corresponds exactly to a local $T_G$8-homotopy between $T_G$9 and $G$0 on the same $G$1-invariant open set (Daundkar et al., 28 Aug 2025).

In the lifting-category framework, equivariant homotopic distance is the lifting category computed in $G$2:

$G$3

with $G$4 or $G$5, both $G$6-maps $G$7 in $G$8. The path-space model and the diagonal model agree:

$G$9

If $G$0 is the equivariant pullback of $G$1 along $G$2, then

$G$3

This places equivariant homotopic distance inside the same formal mechanism that also contains sectional category and topological complexity (Doeraene, 10 Mar 2025).

The same invariant admits equivariant Whitehead–Ganea characterizations. Build the equivariant Whitehead fat wedges $G$4 with structure maps $G$5, and the equivariant Ganea fibrations $G$6. Then

$G$7

and equivalently

$G$8

When the domain is normal, the open-cover definition agrees with these Whitehead–Ganea and lifting-category formulations (Doeraene, 10 Mar 2025).

3. Fundamental properties

Equivariant homotopic distance satisfies symmetry,

$G$9

and has zero value exactly on equivariantly homotopic pairs:

$G$0

It is homotopy invariant on source and target maps: if $G$1 and $G$2, then $G$3. In the lifting-category language this is Axiom A1, homotopy invariance (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

It is also monotone under composition. If $G$4 and $G$5 are $G$6-maps, then

$G$7

If $G$8 has a left $G$9-homotopy inverse, then $G$00; dually, if pre-composition is by a map with a right $G$01-homotopy inverse, then equality also holds. In the axiomatic treatment these are the equivariant forms of the naturality and monotonicity statements under composition and homotopy pullback (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

Product behavior is controlled by subadditivity. For $G$02-maps $G$03 and $G$04-maps $G$05, one has

$G$06

under the stated normality or metrizability hypotheses. There is also the identity

$G$07

in the non-equivariant framework, and the same identities carry over equivariantly when products carry product actions and all maps are $G$08-equivariant (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

A triangular inequality is available under separation hypotheses. One theorem states that if $G$09 is Hausdorff and $G$10 is completely normal, for example if $G$11 is metrizable, then for any $G$12-maps $G$13,

$G$14

In the open-cover-based treatment, the same proof goes through equivariantly with $G$15-invariant covers and $G$16-homotopies under normality of the domain. By contrast, in the classical theory the triangular inequality may fail for non-normal spaces, and a finite $G$17-spaces example is given (Daundkar et al., 28 Aug 2025, Doeraene, 10 Mar 2025, Macías-Virgós et al., 2018).

4. Relation to equivariant Lusternik–Schnirelmann category and equivariant topological complexity

A $G$18-invariant open set $G$19 is $G$20-categorical if the inclusion $G$21 is $G$22-homotopic to a $G$23-map whose image lies in a single orbit. The equivariant LS category $G$24 is the least $G$25 such that $G$26 is covered by $G$27 $G$28-categorical open sets. Equivariant topological complexity is defined by

$G$29

where $G$30 is the free path fibration with diagonal $G$31-action (Daundkar et al., 28 Aug 2025).

Equivariant homotopic distance unifies these invariants. For the projections $G$32 with diagonal $G$33-action,

$G$34

If $G$35 is $G$36-connected and $G$37, choose $G$38 and let $G$39 be constant at $G$40. Then

$G$41

where $G$42 and $G$43. In the lifting-category notation,

$G$44

and

$G$45

Thus sectional category, topological complexity, and homotopic distance are special cases of lifting category (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

Several comparison inequalities follow. For any $G$46-maps $G$47,

$G$48

If $G$49 is $G$50-connected and $G$51, then

$G$52

Equivalently, in the axiomatic notation,

$G$53

These bounds recover the familiar pattern

$G$54

for the diagonal $G$55-action on $G$56 (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

5. Cohomological and dimension-theoretic bounds

Let $G$57 denote Borel equivariant cohomology with coefficients in a commutative ring $G$58, and denote homotopy orbit maps by $G$59. A cohomological lower bound is obtained from the ideal

$G$60

Then

$G$61

the nilpotency of the ideal $G$62. More generally, if there exist classes $G$63 with $G$64 for all $G$65, and $G$66, then

$G$67

For the projections $G$68, with $G$69 a field,

$G$70

so one recovers the zero-divisors cup-length lower bound for $G$71 (Daundkar et al., 28 Aug 2025).

An orbit-space lower bound is also available. If $G$72 is compact Hausdorff and $G$73 is the induced map on orbit spaces in the pullback square, then

$G$74

This compares equivariant homotopic distance with ordinary sectional category on orbit spaces (Daundkar et al., 28 Aug 2025).

The axiomatic framework notes that the paper itself does not develop cohomological lower bounds, but that the lifting-category formulation is compatible with standard cohomological methods. In the equivariant setting one typically works with Borel equivariant cohomology $G$75, and under appropriate finiteness and hypothesis conditions the equivariant zero-divisors cup-length provides a lower bound for $G$76, with further refinements phrased via the nilpotency of the kernel of $G$77 in $G$78 (Doeraene, 10 Mar 2025).

There are also dimension-theoretic upper bounds. If $G$79 is a $G$80-CW complex of dimension at least $G$81, and $G$82 is $G$83–$G$84-connected, meaning that $G$85 is $G$86-connected for every closed subgroup $G$87, then

$G$88

where $G$89 is the minimal $G$90-CW dimension among spaces $G$91-homotopy equivalent to $G$92 (Daundkar et al., 28 Aug 2025).

The classical theory already contained the nonequivariant cup-length estimate. For a unitary commutative ring $G$93, if $G$94, then the cup-length $G$95 satisfies

$G$96

and a homotopy-weight refinement strengthens this bound. The equivariant results are direct analogues in Borel equivariant cohomology (Macías-Virgós et al., 2018).

6. Special cases, examples, and geometric consequences

When $G$97 acts trivially on the relevant spaces, all equivariant notions reduce to classical ones:

$G$98

More generally,

$G$99

If $G$00 acts freely and properly on $G$01 and $G$02, $G$03 are $G$04-maps, and $G$05 are the induced maps, then under the stated hypotheses one has

$G$06

Without freeness and properness, one still has the monotonic bound

$G$07

If $G$08 and one restricts to fixed-point sets, then

$G$09

There is also subgroup comparison: if $G$10 are closed subgroups and the restrictions $G$11 are $G$12-maps, then

$G$13

In particular,

$G$14

and hence

$G$15

(Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

For classical actions, the diagonal and path-fibration models remain equivariant. For $G$16 with the antipodal $G$17-action, the diagonal is $G$18-equivariant, and for $G$19-equivariant maps $G$20 one has

$G$21

where $G$22 is the equivariant pullback of $G$23 along $G$24. Specific values depend on cohomological constraints in $G$25. For tori with translation actions, the same bounds hold, and trivial actions recover the classical invariants (Doeraene, 10 Mar 2025).

A major class of examples is provided by Hopf $G$26-spaces. A pointed $G$27-space is a $G$28-space $G$29 with basepoint $G$30. A Hopf $G$31-space is a pointed $G$32-space with a pointed $G$33-map $G$34 such that $G$35 and $G$36. A division $G$37-map is a pointed $G$38-map $G$39 satisfying

$G$40

If $G$41 is a $G$42-connected Hopf $G$43-space admitting a division $G$44-map, then for pointed $G$45-maps $G$46,

$G$47

and consequently

$G$48

This is the equivariant extension of the Lupton–Scherer phenomenon (Daundkar et al., 28 Aug 2025).

Sharp or near-sharp estimates are known for Hopf $G$49-spheres. For $G$50 with $G$51 acting by complex conjugation, $G$52, so $G$53 is not $G$54-connected and $G$55. For $G$56 with $G$57 acting by automorphisms of the quaternions, $G$58, so again $G$59. By contrast, if $G$60 fixes a nonzero line in $G$61, then $G$62 is $G$63-connected and

$G$64

For any finite subgroup $G$65,

$G$66

Similarly, for $G$67 with $G$68 acting via octonion automorphisms,

$G$69

and for any finite subgroup $G$70,

$G$71

(Daundkar et al., 28 Aug 2025).

Equivariant fibrations provide another geometric application. If $G$72 and $G$73 are $G$74-fibrations with $G$75 $G$76-connected, and if $G$77 and $G$78 satisfy $G$79 and $G$80, then with notation as in the fibrewise theorem,

$G$81

As consequences,

$G$82

and

$G$83

These inequalities are equivariant analogues of the Varadarajan and Farber–Grant inequalities (Daundkar et al., 28 Aug 2025).

7. Assumptions, variants, and scope

The axiomatic treatment works in the unpointed category $G$84 and the pointed category $G$85 of well-pointed spaces, and relies on homotopy-invariant constructions valid in $G$86-categories. Equivariantly, one works in $G$87 and $G$88, provided equivariant homotopy pullbacks, pushouts, and fibrations are used. In the direct development of $G$89, the standing hypotheses are compact Lie groups $G$90, compactly generated Hausdorff $G$91-spaces, and $G$92-CW complexes when equivariant Whitehead and homotopy lifting arguments are invoked (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).

The agreement of the open-cover definition with the Ganea, Whitehead, and lifting-category definitions requires a normality hypothesis on the domain. Equivariantly, the underlying space must be normal, the cover must be $G$93-invariant, and the local homotopies must be $G$94-equivariant. In the pointed category of well-pointed $G$95-spaces, the open-cover characterization uses $G$96-invariant open sets containing the basepoint and pointed $G$97-homotopies; for $G$98, the pointed and unpointed lifting categories agree (Doeraene, 10 Mar 2025).

The diagonal and the path fibration are interchangeable targets for lifting. The equivariant diagonal $G$99 and its fibration replacement $D_G(f,g)$00 are $D_G(f,g)$01-homotopy equivalent as targets for lifting, so either may be used to define $D_G(f,g)$02 and $D_G(f,g)$03. This flexibility is one reason the invariant fits naturally into the larger lifting-category framework (Doeraene, 10 Mar 2025).

The scope of the theory is broad but not unrestricted. Some inequalities require normality or metrizability in order to invoke subadditivity of equivariant sectional category. Borel equivariant cohomology provides effective lower bounds, but computations can be intricate. Extending the compact-Lie-group framework to more general topological groups is possible in the axiomatic setting, where $D_G(f,g)$04 is allowed to be a topological group and the ambient category is $D_G(f,g)$05, but additional care is required for universal constructions and equivariant CW methods. Within these hypotheses, equivariant homotopic distance provides a unified formalism for comparing equivariant maps, for recovering $D_G(f,g)$06 and $D_G(f,g)$07, and for transferring methods between open-cover, sectional-category, and lifting-theoretic approaches (Doeraene, 10 Mar 2025, Daundkar et al., 28 Aug 2025).