Balanced shellings on combinatorial manifolds Martina - PowerPoint PPT Presentation

Balanced shellings on combinatorial manifolds Martina Juhnke-Kubitzke (joint work with Lorenzo Venturello) Einstein Workshop on Discrete Geometry and Topology 2018, Berlin March 14, 2018 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 1 / 21

Balanced combinatorial manifolds 1 Moves on simplicial complexes 2 A balanced analog of Pachner’s theorem for manifolds with boundary 3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 2 / 21

Combinatorial manifolds ∆ connected simplicial complex of dimension d . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds ∆ connected simplicial complex of dimension d . ∆ is a combinatorial d -sphere if it is PL homeomorphic to the boundary of the ( d + 1)-simplex. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds ∆ connected simplicial complex of dimension d . ∆ is a combinatorial d -sphere if it is PL homeomorphic to the boundary of the ( d + 1)-simplex. ∆ is a combinatorial d -manifold without boundary if all its vertex links are combinatorial ( d − 1)-spheres. The link of F ∈ ∆ is lk ∆ ( F ) := { G ∈ ∆ : G ∪ F ∈ ∆ , G ∩ F = ∅} . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds ∆ connected simplicial complex of dimension d . ∆ is a combinatorial d -sphere/ d -ball if it is PL homeomorphic to the boundary of the ( d + 1)-simplex/ d -simplex. ∆ is a combinatorial d -manifold without boundary if all its vertex links are combinatorial ( d − 1)-spheres. The link of F ∈ ∆ is lk ∆ ( F ) := { G ∈ ∆ : G ∪ F ∈ ∆ , G ∩ F = ∅} . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Combinatorial manifolds ∆ connected simplicial complex of dimension d . ∆ is a combinatorial d -sphere/ d -ball if it is PL homeomorphic to the boundary of the ( d + 1)-simplex/ d -simplex. ∆ is a combinatorial d -manifold without boundary if all its vertex links are combinatorial ( d − 1)-spheres. ∆ is a combinatorial d -manifold with boundary if all its vertex links are combinatorial ( d − 1)-spheres or ( d − 1)-balls and its boundary is ∂ ∆ := { F ∈ ∆ : lk ∆ ( F ) is a combinatorial ( d − | F | )-ball } ∪ {∅} . The link of F ∈ ∆ is lk ∆ ( F ) := { G ∈ ∆ : G ∪ F ∈ ∆ , G ∩ F = ∅} . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 3 / 21

Balanced simplicial complexes A simplicial complex ∆ on vertex set V (∆) is properly m -colorable, if Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

Balanced simplicial complexes A simplicial complex ∆ on vertex set V (∆) is properly m -colorable, if the 1-skeleton of ∆ is m -colorable. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

Balanced simplicial complexes A simplicial complex ∆ on vertex set V (∆) is properly m -colorable, if the 1-skeleton of ∆ is m -colorable. ⇔ There exists a map (coloring) φ : V (∆) → { 0 , 1 , . . . , m − 1 } , such that φ ( i ) � = φ ( j ) for all { i , j } ∈ ∆. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

Balanced simplicial complexes A simplicial complex ∆ on vertex set V (∆) is properly m -colorable, if the 1-skeleton of ∆ is m -colorable. ⇔ There exists a map (coloring) φ : V (∆) → { 0 , 1 , . . . , m − 1 } , such that φ ( i ) � = φ ( j ) for all { i , j } ∈ ∆. ∆ is balanced if it is properly (dim ∆ + 1)-colorable. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 4 / 21

The (boundary) of the d -simplex Let σ d be the d -simplex. 4 1 2 3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 5 / 21

The (boundary) of the d -simplex Let σ d be the d -simplex. As the 1-skeleton of σ d is a complete graph on d + 1 vertices, a proper coloring uses at least d + 1 colors. 4 1 2 3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 5 / 21

The (boundary) of the d -simplex Let σ d be the d -simplex. As the 1-skeleton of σ d is a complete graph on d + 1 vertices, a proper coloring uses at least d + 1 colors. ⇒ σ d is balanced, whereas its boundary ∂σ d is not balanced. 4 1 2 3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 5 / 21

The boundary of the ( d + 1)-dimensional cross-polytope Let C d be the boundary of the ( d + 1)-dimensional cross-polytope: C d = { v 0 , w 0 } ∗ · · · ∗ { v d , w d } . w 2 v 0 v 1 w 1 w 0 v 2 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 6 / 21

The boundary of the ( d + 1)-dimensional cross-polytope Let C d be the boundary of the ( d + 1)-dimensional cross-polytope: C d = { v 0 , w 0 } ∗ · · · ∗ { v d , w d } . A ( d + 1)-coloring φ is given by setting φ ( v i ) = φ ( w i ) = i for 0 ≤ i ≤ d . w 2 v 0 v 1 w 1 w 0 v 2 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 6 / 21

The boundary of the ( d + 1)-dimensional cross-polytope Let C d be the boundary of the ( d + 1)-dimensional cross-polytope: C d = { v 0 , w 0 } ∗ · · · ∗ { v d , w d } . A ( d + 1)-coloring φ is given by setting φ ( v i ) = φ ( w i ) = i for 0 ≤ i ≤ d . w 2 v 0 v 1 w 1 w 0 ⇒ C d is balanced. v 2 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 6 / 21

Balanced combinatorial manifolds 1 Moves on simplicial complexes 2 A balanced analog of Pachner’s theorem for manifolds with boundary 3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 7 / 21

Stellar moves and bistellar moves ∆ d -dimensional simplicial complex. The stellar subdivision of ∆ at F ∈ ∆ is sd F (∆) = (∆ \ F ) ∪ ( v ∗ ∂ F ∗ lk ∆ ( F )) . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 8 / 21

Stellar moves and bistellar moves ∆ d -dimensional simplicial complex. The stellar subdivision of ∆ at F ∈ ∆ is sd F (∆) = (∆ \ F ) ∪ ( v ∗ ∂ F ∗ lk ∆ ( F )) . A bistellar move replaces an induced subcomplex A ⊆ ∆ that is isomorphic to a d -dimensional subcomplex of ∂σ d +1 with its complement: ∆ → (∆ \ A ) ∪ ( ∂σ d +1 \ A ) . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 8 / 21

What about combinatorial manifolds with boundary? Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 9 / 21

Shellings and their inverses ∆ pure d -dimensional simplicial complex. An elementary shelling removes a facet F ∈ ∆ with the property that { G ⊆ F : G / ∈ ∆ \ F } has a unique minimal element. ∆ → ∆ \ F . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 10 / 21

Shellings and their inverses ∆ pure d -dimensional simplicial complex. An elementary shelling removes a facet F ∈ ∆ with the property that { G ⊆ F : G / ∈ ∆ \ F } has a unique minimal element. ∆ → ∆ \ F . The inverse operation is called an inverse shelling. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 10 / 21

Shellings and their inverses ∆ pure d -dimensional simplicial complex. An elementary shelling removes a facet F ∈ ∆ with the property that { G ⊆ F : G / ∈ ∆ \ F } has a unique minimal element. ∆ → ∆ \ F . The inverse operation is called an inverse shelling. A shelling on ∆ corresponds to a bistellar flip on ∂ ∆. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 10 / 21

What about balanced combinatorial manifolds? Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 11 / 21

Cross-flips ∆ balanced d -dimensional simplicial complex. A cross-flip replaces an induced subcomplex D ⊆ ∆ that is isomorphic to a shellable and coshellable subcomplex of C d with its complement: ∆ → (∆ \ D ) ∪ ( C d \ D ) . Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 12 / 21

Cross-flips ∆ balanced d -dimensional simplicial complex. A cross-flip replaces an induced subcomplex D ⊆ ∆ that is isomorphic to a shellable and coshellable subcomplex of C d with its complement: ∆ → (∆ \ D ) ∪ ( C d \ D ) . Cross-flips preserve balancedness. Cross-flips preserve the PL homeomorphism type. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 12 / 21

Cross-flips in dimension 2 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 13 / 21

What about balanced combinatorial manifolds with boundary? Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 14 / 21

Balanced combinatorial manifolds 1 Moves on simplicial complexes 2 A balanced analog of Pachner’s theorem for manifolds with boundary 3 Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 15 / 21

The main result Theorem (J.-K., Venturello; 2018+) Balanced combinatorial manifolds with boundary are PL homeo- morphic if and only if they are connected by a sequence of shellings and inverse shellings preserving balancedness in each step. Martina Juhnke-Kubitzke Balanced shellings on manifolds March 14, 2018 16 / 21