The Coherent Framed Join and Biassociahedra Joint work with Samson - - PowerPoint PPT Presentation
The Coherent Framed Join and Biassociahedra Joint work with Samson Saneblidze Ron Umble Millersville University Celebrating the legacies of Jim Stasheff and Murray Gerstenhaber 5 March 2018 Background In our 2011 paper entitled,
Joint work with Samson Saneblidze Ron Umble Millersville University
Celebrating the legacies of Jim Stasheff and Murray Gerstenhaber
5 March 2018
In our 2011 paper entitled, “Matrads, Biassociahedra, and
A∞-bialgebras”, we constructed a basis for the free matrad H∞ and the polytopes KKn,m in the ranges 1 ≤ m ≤ 3 and 1 ≤ n ≤ 3
In our 2011 paper entitled, “Matrads, Biassociahedra, and
A∞-bialgebras”, we constructed a basis for the free matrad H∞ and the polytopes KKn,m in the ranges 1 ≤ m ≤ 3 and 1 ≤ n ≤ 3
Outside these ranges we are unable to define an operator that
simultaneously preserves coherency and satisfies d2 = 0. In fact...
In our 2011 paper entitled, “Matrads, Biassociahedra, and
A∞-bialgebras”, we constructed a basis for the free matrad H∞ and the polytopes KKn,m in the ranges 1 ≤ m ≤ 3 and 1 ≤ n ≤ 3
Outside these ranges we are unable to define an operator that
simultaneously preserves coherency and satisfies d2 = 0. In fact...
When m = n = 4, Saneblidze constructed an example with
the following property: If we use all available components of the face operator to extend the differential, coherency is lost; if we use only those available components that preserve coherency, d2 = 0
In our 2011 paper entitled, “Matrads, Biassociahedra, and
A∞-bialgebras”, we constructed a basis for the free matrad H∞ and the polytopes KKn,m in the ranges 1 ≤ m ≤ 3 and 1 ≤ n ≤ 3
Outside these ranges we are unable to define an operator that
simultaneously preserves coherency and satisfies d2 = 0. In fact...
When m = n = 4, Saneblidze constructed an example with
the following property: If we use all available components of the face operator to extend the differential, coherency is lost; if we use only those available components that preserve coherency, d2 = 0
Let us construct KKn,m in the ranges 1 ≤ m ≤ 3 and
1 ≤ n ≤ 3
An ordered set is ∅ or a finite strictly increasing subset of N
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0 P r (A) denotes the augmented length r partitions of A
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0 P r (A) denotes the augmented length r partitions of A
P
r (∅) = {0| · · · |0} with r empty blocks
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0 P r (A) denotes the augmented length r partitions of A
P
r (∅) = {0| · · · |0} with r empty blocks
P
r (A) = {A1| · · · |Ar } , where A1 ∪ · · · ∪ Ar = A
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0 P r (A) denotes the augmented length r partitions of A
P
r (∅) = {0| · · · |0} with r empty blocks
P
r (A) = {A1| · · · |Ar } , where A1 ∪ · · · ∪ Ar = A
Some Ai may be empty
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0 P r (A) denotes the augmented length r partitions of A
P
r (∅) = {0| · · · |0} with r empty blocks
P
r (A) = {A1| · · · |Ar } , where A1 ∪ · · · ∪ Ar = A
Some Ai may be empty
Define π : P (A) → P (A) by deleting empty blocks
An ordered set is ∅ or a finite strictly increasing subset of N Let A be an ordered set; let r > 0 P r (A) denotes the augmented length r partitions of A
P
r (∅) = {0| · · · |0} with r empty blocks
P
r (A) = {A1| · · · |Ar } , where A1 ∪ · · · ∪ Ar = A
Some Ai may be empty
Define π : P (A) → P (A) by deleting empty blocks Dimension
A bipartition is a pair β α := (α, β) ∈ P r (A) × P r (B)
A bipartition is a pair β α := (α, β) ∈ P r (A) × P r (B) Example
7|0|6 0|1|0 ∈ P
3 ({1}) × P 3 ({6, 7})
A bipartition is a pair β α := (α, β) ∈ P r (A) × P r (B) Example
7|0|6 0|1|0 ∈ P
3 ({1}) × P 3 ({6, 7}) Let a1, . . . , ap, b1, . . . , bq be ordered sets; R = (rij) ∈ Nq×p
A bipartition is a pair β α := (α, β) ∈ P r (A) × P r (B) Example
7|0|6 0|1|0 ∈ P
3 ({1}) × P 3 ({6, 7}) Let a1, . . . , ap, b1, . . . , bq be ordered sets; R = (rij) ∈ Nq×p For simplicity, assume the ai’s (and bj’s) are disjoint
A bipartition is a pair β α := (α, β) ∈ P r (A) × P r (B) Example
7|0|6 0|1|0 ∈ P
3 ({1}) × P 3 ({6, 7}) Let a1, . . . , ap, b1, . . . , bq be ordered sets; R = (rij) ∈ Nq×p For simplicity, assume the ai’s (and bj’s) are disjoint Choose bipartitions
βij αij ∈ P
rij (aj) × P rij (bi)
A bipartition is a pair β α := (α, β) ∈ P r (A) × P r (B) Example
7|0|6 0|1|0 ∈ P
3 ({1}) × P 3 ({6, 7}) Let a1, . . . , ap, b1, . . . , bq be ordered sets; R = (rij) ∈ Nq×p For simplicity, assume the ai’s (and bj’s) are disjoint Choose bipartitions
βij αij ∈ P
rij (aj) × P rij (bi)
αij
q×p is a bipartition matrix over {ai, bj} w.r.t. R
Example
4|5 1|0 5|4 3|2 7|0|6 0|1|0 67 23
is a bipartition matrix
with respect to
2 3 1
Let λ be an ordered subset of {1, 2, . . . , n} of order k
Let λ be an ordered subset of {1, 2, . . . , n} of order k µλ : P n+1 (A) → P k+1 (A)
Let λ be an ordered subset of {1, 2, . . . , n} of order k µλ : P n+1 (A) → P k+1 (A) Example: µ{2,3,5}2|1|0|5|4|3 = 12|0|45|3
Let λ be an ordered subset of {1, 2, . . . , n} of order k µλ : P n+1 (A) → P k+1 (A) Example: µ{2,3,5}2|1|0|5|4|3 = 12|0|45|3 Extreme cases:
µ∅ (A1| · · · |An+1) = A µ{1,2,...,n} (A1| · · · |An+1) = A1| · · · |An+1
Given a q × p bipartition matrix βij
αij
there is a unique q × p matrix of ordered sets (λij) such that
(equal denominators in jth column)
Given a q × p bipartition matrix βij
αij
there is a unique q × p matrix of ordered sets (λij) such that
(equal denominators in jth column)
(equal numerators in ith row)
Given a q × p bipartition matrix βij
αij
there is a unique q × p matrix of ordered sets (λij) such that
(equal denominators in jth column)
(equal numerators in ith row)
Given a q × p bipartition matrix βij
αij
there is a unique q × p matrix of ordered sets (λij) such that
(equal denominators in jth column)
(equal numerators in ith row)
Example:
45|0 1|0 5|4|0 0|2|3 7|0|0|6 0|1|0|0 0|7|6 2|0|3
µλ
45|0 1|0 45|0 2|3 7|6 1|0 7|6 2|3
, where λ = {1} {2} {2} {2}
Definition A bipartition matrix is indecomposable if its
associated λ matrix is null
Definition A bipartition matrix is indecomposable if its
associated λ matrix is null
Theorem A bipartition matrix has a unique indecomposable
factorization
Let B be an ordered set
Let B be an ordered set ACPBB = B
Let B be an ordered set ACPBB = B ACPB∅ = 0| · · · |0 (# empty blocks = #B + 1)
Let B be an ordered set ACPBB = B ACPB∅ = 0| · · · |0 (# empty blocks = #B + 1) ACP{1,2,...,9} {2, 5, 6, 8} = 0|2|0|56|8|0
Given C = B1| · · · |Br
A1| · · · |Ar , for each k = 1, 2, . . . r :
Given C = B1| · · · |Br
A1| · · · |Ar , for each k = 1, 2, . . . r :
Compute
ak,1| · · · |ak,sk := ACPA1∪···∪Ak Ak bk,1| · · · |bk,tk := ACPBk ∪···∪Br Bk
Given C = B1| · · · |Br
A1| · · · |Ar , for each k = 1, 2, . . . r :
Compute
ak,1| · · · |ak,sk := ACPA1∪···∪Ak Ak bk,1| · · · |bk,tk := ACPBk ∪···∪Br Bk
Construct the bipartition matrix
Ck =
bk,1 ak,1
· · ·
bk,1 ak,sk
. . . . . .
bk,tk ak,1
· · ·
bk,tk ak,sk
Given C = B1| · · · |Br
A1| · · · |Ar , for each k = 1, 2, . . . r :
Compute
ak,1| · · · |ak,sk := ACPA1∪···∪Ak Ak bk,1| · · · |bk,tk := ACPBk ∪···∪Br Bk
Construct the bipartition matrix
Ck =
bk,1 ak,1
· · ·
bk,1 ak,sk
. . . . . .
bk,tk ak,1
· · ·
bk,tk ak,sk
C = C1 · · · Cr
Example
56|7|8 1|23|4 1 = ACP11 56|0|0 = ACP567856 0|23 = ACP12323 7|0 = ACP787 0|0|0|4 = ACP12344 8 = ACP88
Example
56|7|8 1|23|4 1 = ACP11 56|0|0 = ACP567856 0|23 = ACP12323 7|0 = ACP787 0|0|0|4 = ACP12344 8 = ACP88
1|23|4 =
56 1 1 1
7
7 23 23
8 8 8 4
A ←
#B+1 #A+1
A ←
#B+1 #A+1
1|23|4
56 1 1 1
7
7 23 23
8 8 8 4
=
A null matrix with entries of the form 0|···|0 0|···|0 has dim 0
A null matrix with entries of the form 0|···|0 0|···|0 has dim 0
B
A
A null matrix with entries of the form 0|···|0 0|···|0 has dim 0
B
A
|C1 · · · Cr| := |C1| + · · · + |Cr|
A null matrix with entries of the form 0|···|0 0|···|0 has dim 0
B
A
|C1 · · · Cr| := |C1| + · · · + |Cr| Unique factorization ⇒ Define |C| for C indecomposable
A null matrix with entries of the form 0|···|0 0|···|0 has dim 0
B
A
|C1 · · · Cr| := |C1| + · · · + |Cr| Unique factorization ⇒ Define |C| for C indecomposable Let C =
βij
αij
If βij αij = 0|···|0 αij
for all (i, j) , let Ci∗ denote the ith row of C
If βij αij = 0|···|0 αij
for all (i, j) , let Ci∗ denote the ith row of C
Let (λi 1 · · · λi p) be the λ matrix associated with Ci∗
If βij αij = 0|···|0 αij
for all (i, j) , let Ci∗ denote the ith row of C
Let (λi 1 · · · λi p) be the λ matrix associated with Ci∗ Define
A1| · · · |An A
1| · · · |A n :=
1
| · · · |
n
If βij αij = 0|···|0 αij
for all (i, j) , let Ci∗ denote the ith row of C
Let (λi 1 · · · λi p) be the λ matrix associated with Ci∗ Define
A1| · · · |An A
1| · · · |A n :=
1
| · · · |
n
∧
αi := µλi
1(αi1) · · · µλi p(αip))
If βij αij = 0|···|0 αij
for all (i, j) , let Ci∗ denote the ith row of C
Let (λi 1 · · · λi p) be the λ matrix associated with Ci∗ Define
A1| · · · |An A
1| · · · |A n :=
1
| · · · |
n
∧
αi := µλi
1(αi1) · · · µλi p(αip))
Define |C| :=
∑
1≤i≤q
|
∧
αi|
Example
3
Example
3
|C| is not necessarily the sum of the dim’s of its entries
Example
3
|C| is not necessarily the sum of the dim’s of its entries If cij = βij 0|···|0 for all (i, j) , form partitions ∨
βj in each column
Example
3
|C| is not necessarily the sum of the dim’s of its entries If cij = βij 0|···|0 for all (i, j) , form partitions ∨
βj in each column
Define |C| =
∑
1≤j≤p
|
∨
βj|
Example
3
|C| is not necessarily the sum of the dim’s of its entries If cij = βij 0|···|0 for all (i, j) , form partitions ∨
βj in each column
Define |C| =
∑
1≤j≤p
|
∨
βj|
Otherwise...
Deleting or inserting empty blocks in an entry of a bipartition
matrix may preserve or change dimension
Deleting or inserting empty blocks in an entry of a bipartition
matrix may preserve or change dimension
Discard bipartition matrices whose dimension increases when
empty blocks are inserted
Deleting or inserting empty blocks in an entry of a bipartition
matrix may preserve or change dimension
Discard bipartition matrices whose dimension increases when
empty blocks are inserted
Example
Discard the 1-dim’l indecomposable matrix C = 0|1 1|0 0|1 1|0 1 1
the 3-dim’l decomposable 0|1 1|0 0|1 1|0 0|1 0|1
1 1 1 1
1 1 1 1 1
Equate bipartition matrices of the same dimension that differ
Equate bipartition matrices of the same dimension that differ
Example
1 3 0|0|0 0|1|0 0|0|0 0|0|3
=
1 3 0|0 1|0 0|0 0|3
Equate bipartition matrices of the same dimension that differ
Example
1 3 0|0|0 0|1|0 0|0|0 0|0|3
=
1 3 0|0 1|0 0|0 0|3
Only preserve empty blocks necessary to preserve dimension
Equate bipartition matrices of the same dimension that differ
Example
1 3 0|0|0 0|1|0 0|0|0 0|0|3
=
1 3 0|0 1|0 0|0 0|3
Only preserve empty blocks necessary to preserve dimension Example
Preserve all empty blocks in C =
1 3 0|0 1|0 0|0 0|3
Removing empty blocks in the second row increases dimension
Definition
A q × p indecomposable bipartition matrix βij
αij
Definition
A q × p indecomposable bipartition matrix βij
αij
column coherent if
π(
∧
αq) × · · · × π(
∧
α1) ∆(q−1)(P#(a1∪···∪ap))
Definition
A q × p indecomposable bipartition matrix βij
αij
column coherent if
π(
∧
αq) × · · · × π(
∧
α1) ∆(q−1)(P#(a1∪···∪ap))
row coherent if
π(
∨
β1) × · · · × π(
∨
βp) ∆(p−1)(P#(b1∪···∪bq))
Definition
A q × p indecomposable bipartition matrix βij
αij
column coherent if
π(
∧
αq) × · · · × π(
∧
α1) ∆(q−1)(P#(a1∪···∪ap))
row coherent if
π(
∨
β1) × · · · × π(
∨
βp) ∆(p−1)(P#(b1∪···∪bq))
coherent if column and row coherent
Given a(m) and b(n) of orders m and n, and r ≥ 1, let
β α ∈ P
r(a(m)) × P r(b(n))
Given a(m) and b(n) of orders m and n, and r ≥ 1, let
β α ∈ P
r(a(m)) × P r(b(n)) If r = 1 or mn = 0, the set of coherent framed elements
α c β := β α
Given a(m) and b(n) of orders m and n, and r ≥ 1, let
β α ∈ P
r(a(m)) × P r(b(n)) If r = 1 or mn = 0, the set of coherent framed elements
α c β := β α
elements α c β has been defined for all
β α ∈ P(a(s)) × P(b(t)) such that (s, t) ≤ (m, n) and
s + t < m + n
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Compute a1| · · · |ap := ACPA1∪···∪Ak Ak
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Compute a1| · · · |ap := ACPA1∪···∪Ak Ak Compute b1| · · · |bq := ACPBk ∪···∪Br Bk
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Compute a1| · · · |ap := ACPA1∪···∪Ak Ak Compute b1| · · · |bq := ACPBk ∪···∪Br Bk Choose R ∈ Nq×p and indecomposable
i
α
j
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Compute a1| · · · |ap := ACPA1∪···∪Ak Ak Compute b1| · · · |bq := ACPBk ∪···∪Br Bk Choose R ∈ Nq×p and indecomposable
i
α
j
Choose ck
ij ∈ α j c β i
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Compute a1| · · · |ap := ACPA1∪···∪Ak Ak Compute b1| · · · |bq := ACPBk ∪···∪Br Bk Choose R ∈ Nq×p and indecomposable
i
α
j
Choose ck
ij ∈ α j c β i
Form the coherent framed matrix Ck =
ij
Given B1| · · · |Br
A1| · · · |Ar = β α, for k = 1, 2, . . . , r :
Compute a1| · · · |ap := ACPA1∪···∪Ak Ak Compute b1| · · · |bq := ACPBk ∪···∪Br Bk Choose R ∈ Nq×p and indecomposable
i
α
j
Choose ck
ij ∈ α j c β i
Form the coherent framed matrix Ck =
ij
α c β := {C1 · · · Cr} , where Ci ranges over all possible coherent framed matrices and the product is formal juxtaposition
Definition
The coherent framed join of a(m) and b(n) is the set a(m) pp b(n) :=
α ∈P r (a(m))×P r (b(n))
r≥1
α c β
Definition
The coherent framed join of a(m) and b(n) is the set a(m) pp b(n) :=
α ∈P r (a(m))×P r (b(n))
r≥1
α c β
Example
1 pp 1 =
1, 0|1 1|0 =
1 1
1
1
1|0 0|1 =
1
1
Definition
The coherent framed join of a(m) and b(n) is the set a(m) pp b(n) :=
α ∈P r (a(m))×P r (b(n))
r≥1
α c β
Example
1 pp 1 =
1, 0|1 1|0 =
1 1
1
1
1|0 0|1 =
1
1
0|1 1|0 1 1 1|0 0|1
Definition
The coherent framed join of a(m) and b(n) is the set a(m) pp b(n) :=
α ∈P r (a(m))×P r (b(n))
r≥1
α c β
Example
1 pp 1 =
1, 0|1 1|0 =
1 1
1
1
1|0 0|1 =
1
1
0|1 1|0 1 1 1|0 0|1
The Hopf relation holds up to homotopy
Example
For 12 pp 1, let W be the set obtained by inserting empty blocks into
1 12 in all possible ways that
preserve coherence
Example
For 12 pp 1, let W be the set obtained by inserting empty blocks into
1 12 in all possible ways that
preserve coherence
W =
12, 0|1 1|2, 0|1 2|1, 1|0 1|2, 1|0 2|1, 1|0 0|12, 0|0|1 1|2|0, 0|0|1 2|1|0, 0|1|0 1|0|2, 0|1|0 2|0|1, 1|0|0 0|1|2, 1|0|0 0|2|1
Example
For 12 pp 1, let W be the set obtained by inserting empty blocks into
1 12 in all possible ways that
preserve coherence
W =
12, 0|1 1|2, 0|1 2|1, 1|0 1|2, 1|0 2|1, 1|0 0|12, 0|0|1 1|2|0, 0|0|1 2|1|0, 0|1|0 1|0|2, 0|1|0 2|0|1, 1|0|0 0|1|2, 1|0|0 0|2|1
0|1 12|0 =
12 12
1
1 1
π(
∧
α2) × π(
∧
α1) = 12 × 12 ∆(1)(P2)
Example
For 12 pp 1, let W be the set obtained by inserting empty blocks into
1 12 in all possible ways that
preserve coherence
W =
12, 0|1 1|2, 0|1 2|1, 1|0 1|2, 1|0 2|1, 1|0 0|12, 0|0|1 1|2|0, 0|0|1 2|1|0, 0|1|0 1|0|2, 0|1|0 2|0|1, 1|0|0 0|1|2, 1|0|0 0|2|1
0|1 12|0 =
12 12
1
1 1
π(
∧
α2) × π(
∧
α1) = 12 × 12 ∆(1)(P2)
Replace entries in all possible ways to obtain coherence
12|0 c 0|1 = 0|0
2|1 12
1
1 1
0|0 1|2
1
1 1
0|0
2|1 0|0 1|2
1
1 1
Example
For 12 pp 1, let W be the set obtained by inserting empty blocks into
1 12 in all possible ways that
preserve coherence
W =
12, 0|1 1|2, 0|1 2|1, 1|0 1|2, 1|0 2|1, 1|0 0|12, 0|0|1 1|2|0, 0|0|1 2|1|0, 0|1|0 1|0|2, 0|1|0 2|0|1, 1|0|0 0|1|2, 1|0|0 0|2|1
0|1 12|0 =
12 12
1
1 1
π(
∧
α2) × π(
∧
α1) = 12 × 12 ∆(1)(P2)
Replace entries in all possible ways to obtain coherence
12|0 c 0|1 = 0|0
2|1 12
1
1 1
0|0 1|2
1
1 1
0|0
2|1 0|0 1|2
1
1 1
Let m = {1, 2, . . . , m} ; let ρ ∈ m pp n
Let m = {1, 2, . . . , m} ; let ρ ∈ m pp n For top dim’l ρ = n m define
˜ ∂ n m
Let m = {1, 2, . . . , m} ; let ρ ∈ m pp n For top dim’l ρ = n m define
˜ ∂ n m
Example
˜ ∂( 1
12) =
1|2, 0|1 2|1, 1|0 1|2, 1|0 2|1, 1|0 0|12,
0|0
2|1 12
1
1 1
0|0 1|2
1
1 1
For lower dim’l cells insert empty blocks and subdivide in all
possible ways that preserve coherence
For lower dim’l cells insert empty blocks and subdivide in all
possible ways that preserve coherence
˜
∂
1|2
1|0|2 ∪ 0|0|1 1|2|0
For lower dim’l cells insert empty blocks and subdivide in all
possible ways that preserve coherence
˜
∂
1|2
1|0|2 ∪ 0|0|1 1|2|0 ˜
∂
0|12
0|1|2 ∪ 1|0|0 0|2|1
For lower dim’l cells insert empty blocks and subdivide in all
possible ways that preserve coherence
˜
∂
1|2
1|0|2 ∪ 0|0|1 1|2|0 ˜
∂
0|12
0|1|2 ∪ 1|0|0 0|2|1 ˜
∂ 0|0
2|1 12
1
1 1
0|0
2|1 0|0 1|2
1
1 1
∪ 0|0
2|1 0|0 2|1
1
1 1
Define an equivalence relation ∼ on a(m) pp b(n):
Define an equivalence relation ∼ on a(m) pp b(n):
C =
∼ C =
ij
ij differ only in the
number or placement of empty blocks 0
Define an equivalence relation ∼ on a(m) pp b(n):
C =
∼ C =
ij
ij differ only in the
number or placement of empty blocks 0
Example 1 3
=
1|0 0|0 0|3
0|1 0|0 3|0
Define an equivalence relation ∼ on a(m) pp b(n):
C =
∼ C =
ij
ij differ only in the
number or placement of empty blocks 0
Example 1 3
=
1|0 0|0 0|3
0|1 0|0 3|0
The reduced coherent framed join of a(m) and b(n) is the set a(m) kk b(n) = a(m) pp b(n)/ ∼
Define an equivalence relation ∼ on a(m) pp b(n):
C =
∼ C =
ij
ij differ only in the
number or placement of empty blocks 0
Example 1 3
=
1|0 0|0 0|3
0|1 0|0 3|0
The reduced coherent framed join of a(m) and b(n) is the set a(m) kk b(n) = a(m) pp b(n)/ ∼
In a(m) kk b(n)
Define an equivalence relation ∼ on a(m) pp b(n):
C =
∼ C =
ij
ij differ only in the
number or placement of empty blocks 0
Example 1 3
=
1|0 0|0 0|3
0|1 0|0 3|0
The reduced coherent framed join of a(m) and b(n) is the set a(m) kk b(n) = a(m) pp b(n)/ ∼
In a(m) kk b(n)
Dim of a matrix is the sum of the dim’s of its entries
Define an equivalence relation ∼ on a(m) pp b(n):
C =
∼ C =
ij
ij differ only in the
number or placement of empty blocks 0
Example 1 3
=
1|0 0|0 0|3
0|1 0|0 3|0
The reduced coherent framed join of a(m) and b(n) is the set a(m) kk b(n) = a(m) pp b(n)/ ∼
In a(m) kk b(n)
Dim of a matrix is the sum of the dim’s of its entries Differential acts on matrix as a derivation of its entries
KKn+1,m+1 ↔ m kk n
KKn+1,m+1 ↔ m kk n In KK1,4 ↔ 3 kk 0 we have
2 1 3
2 0|0 1|0 0|0 0|3
2 0|0 0|1 0|0 3|0
0|0 2|13 = 0|0|0 2|1|3 = 0|0|0 2|3|1
Front view Rear view
∂KK3,3 consists of 8 heptagons and 22 squares
We define a global differential on a(m) pp b(n) but at the
cost of coherence
We define a global differential on a(m) pp b(n) but at the
cost of coherence
Identify the cellular chains C∗ (KK) with the free matrad H∞
We define a global differential on a(m) pp b(n) but at the
cost of coherence
Identify the cellular chains C∗ (KK) with the free matrad H∞ Definition
An A∞-bialgebra is an algebra over H∞
Applications require parallel construction of bimultiplihedra JJ
Applications require parallel construction of bimultiplihedra JJ We transfer a biassociative bialgebra on chains to homology
Applications require parallel construction of bimultiplihedra JJ We transfer a biassociative bialgebra on chains to homology Realize an induced A∞-bialgebra structure on homology
Applications require parallel construction of bimultiplihedra JJ We transfer a biassociative bialgebra on chains to homology Realize an induced A∞-bialgebra structure on homology Theorem
A non-trivial A∞-coalgebra structure on H∗ (X; Q) induces a non-trivial A∞-bialgebra structure on H∗ (ΩΣX; Q)
Applications require parallel construction of bimultiplihedra JJ We transfer a biassociative bialgebra on chains to homology Realize an induced A∞-bialgebra structure on homology Theorem
A non-trivial A∞-coalgebra structure on H∗ (X; Q) induces a non-trivial A∞-bialgebra structure on H∗ (ΩΣX; Q)
The A∞-bialgebra structure on H∗ (ΩΣX; Q) is a rational
homology invariant
Applications require parallel construction of bimultiplihedra JJ We transfer a biassociative bialgebra on chains to homology Realize an induced A∞-bialgebra structure on homology Theorem
A non-trivial A∞-coalgebra structure on H∗ (X; Q) induces a non-trivial A∞-bialgebra structure on H∗ (ΩΣX; Q)
The A∞-bialgebra structure on H∗ (ΩΣX; Q) is a rational
homology invariant
Prior to this work, all known rational homology invariants of
ΩΣX were trivial