Spaces of embeddings: new directions Danica Kosanovi ć Zürich, 10.12.2019
G o a l u n d e r s t a n d s p a c e s o f s m o o t h e m b e d d i n g s ( w i t h W h i t n e y C t o p o l o g y ) co - qub . ( P . M ) " V ' smooth compact manifolds with OP , 0M¥ 0
G o a l u n d e r s t a n d s p a c e s o f s m o o t h e m b e d d i n g s ( w i t h W h i t n e y C t o p o l o g y ) co - V :P # iii. qubfp . M ) Fix smooth compact manifolds with Then = V tap flap
M o t i v a t i o n G o a l Classical knot theory ? !w .fi#=tnotYisotopy Huh u n d e r s t a n d s p a c e s o f s m o o t h e m b e d d i n g s ( w i t h W h i t n e y C t o p o l o g y ) co - qubfp.TN " V " - i. ⇒ Fix V :P # smooth compact manifolds with . Then f- I Hap
M o t i v a t i o n Classical knot theory G o a l - t.EU/5'.Py3l--%aebofI.I4--knotYisotopy u n d e r s t a n d fin "T%sotopy any M 45 ' = s p a c e s o f s m o o t h e m b e d d i n g s of dimensions ( w i t h W h i t n e y C t o p o l o g y ) as Higher-dimensional knots Raub .is?D4l=2-knotYisotopy " Eoubfp , M ) - " ( / " 172,2 ftaefliger ) Haub , # " ) " its ± - Fix V :P ⇐ M smooth compact Knot families manifolds with - * .gg#.s....I.y-.z*.gnon.mn.aefDa.y op ,o* Then - Hap flop Diffeomorphism groups - Ditto M ) - qubdm.my
' - I P My main project concerns the case and the following problem. Conjecture . [Budney-Conant-Koytcheff-Sinha '17] For n>0 the evaluation map . I 3) → I Wn : Embo ( I gives a universal Vassiliev additive knot invariant of type <n.
' = I P My main project concerns the case and the following problem. Conjecture . [Budney-Conant-Koytcheff-Sinha '17] For n>0 the evaluation map . I 3) → I Wn : Embo ( I gives a universal Vassiliev additive knot invariant of type <n. → I - I The tower of spaces ... ... → - n and evaluation maps eun come from the embedding calculus of Goodwillie and Weiss '99. This is a powerful technique for studying Embo ( P . M ) any space of embeddings . By Goodwillie and Klein '15 I if dim(M) - dim(P) > 2 then becomes increasingly better approximation. They predict that for dim(P)=1, dim(M)=3 is surjective on components . own
' - I P My main project concerns the case and the following problem. Conjecture . [Budney-Conant-Koytcheff-Sinha '17] For n>0 the evaluation map Wn : Embo ( I. I3 ) - I gives a universal Vassiliev additive knot invariant of type <n. → I - I The tower of spaces ... ... Theory of finite type invariants by Vassiliev '90. → - n and evaluation maps E.g. all quantum invariants are of finite type. eun come from the embedding calculus of Many questions remain open... Goodwillie and Weiss '99. Geometrically studied by Gusarov '00, Habiro '00, Conant-Teichner '04. This is a powerful technique for studying Eoubfp , M ) The last uses capped grope cobordisms . any space of embeddings . This gives a filtration on the monoid of knots: By Goodwillie and Klein '15 to Emboli .IM ' HI := I if dim(M) - dim(P) > 2 then becomes increasingly better approximation. They predict that for dim(P)=1, dim(M)=3 is surjective on components . em . .
Conjecture . [Budney-Conant-Koytcheff-Sinha '17] For each n>0 the evaluation map . I 3) → I eun : Embo ( I gives a universal additive Vassiliev knot invariant of type <n, meaning that the induced map on path components is a monoid homomorphism which factors as ' ik ! . II ' n Et TOI and the horizontal map is an isomorphism.
Conjecture . [Budney-Conant-Koytcheff-Sinha '17] For each n>0 the evaluation map . I 3) → I Endo , ( I eun : gives a universal additive Vassiliev knot invariant of type <n, meaning that the induced map on path components is a monoid homomorphism which factors as / " ¥ This was proven in BCKS'17. . Et TOI This is an equivalence This is the relation that and the horizontal map is an isomorphism. meaning of uses gropes. ← universality.
Conjecture . [Budney-Conant-Koytcheff-Sinha '17] For each n>0 the evaluation map . I 3) → I eun : Embo ( I gives a universal additive Vassiliev knot invariant of type <n, meaning that the induced map on path components is a monoid homomorphism which factors as 7 " ¥t This was proven in BCKS'17. ' II n E- Io T and the horizontal map is an isomorphism. Theorem [K.] The horizontal map is surjective. This is also one case of the Goodwillie-Klein prediction. It will follow from a more general result we state later.
Definition . Two knots are n-equivalent K ~ K' if there is a finite sequence of h capped grope cobordisms of degree n from K to K'.
Definition . Two knots are n-equivalent K ~ K' if there is a finite sequence of h capped grope cobordisms of degree n from K to K'. A capped grope cobordism of degree n is a certain 2-complex built out of embedded tori and disks and shaped after a rooted planar tree with n labelled leaves .
Definition . Two knots are n-equivalent K ~ K' if there is a finite sequence of h capped grope cobordisms of degree n from K to K'. A capped grope cobordism of degree n is a certain 2-complex built out of embedded tori and disks and shaped after a rooted planar tree with n labelled leaves . Examples. i ⇐ o . ' K . v l ' K ° ⇐ U
Examples continued. The knots and are both isotopic to the trefoil T. Hence: U ~ T and U ~ T. A 2
Examples continued. The knots i. and are both isotopic to the trefoil T. Hence: U ~ T and U ~ T. 1 2 But, one can prove that U ~ T. 13 Mh Moreover, T is the generator of . . Theorem [Gusarov '00, Habiro '00, Conant-Teichner '04] Whn is a finitely generated abelian group.
Examples continued. The knots and are both isotopic to the trefoil T. Hence: U ~ T and U ~ T. 1 2 But, one can prove that U ~ T. 13 Kh Moreover, T is the generator of . . Theorem [Gusarov '00, Habiro '00, Conant-Teichner '04] I is a finitely generated abelian group. n However, this group has not been computed for any n > 7. ÷ ÷ ÷ " ' Is : in ' # I
Theorem [Conant-Teichner '04b] : Atanas I There is a surjective homomorphism of abelian groups . Ren n a : Here is the abelian group generated by rooted planar binary trees with <n leaves modulo certain relations (related to graph complexes).
Theorem [Conant-Teichner '04b] : Atanas I There is a surjective homomorphism of abelian groups . Ren n at Here is the abelian group generated by . rooted planar binary trees with <n leaves modulo certain relations (related to graph complexes). Theorem [Kontsevich '93, Bott-Taubes '94, Altschuler-Freidel '97] Ran has a rational inverse Atan ④ Q I ④ IQ 2. n : n (This is a universal additive Vassiliev invariant over Q.) I
Theorem [Conant-Teichner '04b] : Atanas I There is a surjective homomorphism of abelian groups . Ren n a : Here is the abelian group generated by rooted planar binary trees with <n leaves modulo certain relations (related to graph complexes). Theorem [Kontsevich '93, Bott-Taubes '94, Altschuler-Freidel '97] Ren has a rational inverse Atan ④ Q I ④ IQ 2. n : n (This is a universal additive Vassiliev invariant over Q.) I Open Problems. t - Are there any torsion elements in ? n a : - Are there any torsion elements in ? : At Eos I Ron - Is it true that ? n
Theorem [K.] The following diagram commutes: t Ann d is given by higher differentials in the Ray y Bousfield-Kan spectral d sequence. B. Am B. ikyn Io T
Theorem [K.] The following diagram commutes: t Ann d is given by higher differentials in the Ray \ Bousfield-Kan spectral d sequence. B.B. A. ikyn TOI In particular, this proves one half of the Conjecture: is surjective. et n
Theorem [K.] The following diagram commutes: t Ann d is given by higher differentials in the Ray \ Bousfield-Kan spectral d sequence. B. Am b. ikyn TOI In particular, this proves one half of the Conjecture: is surjective. ein Corollaries . [K] 1. If d is injective over some coefficient group A (i.e. the spectral sequence over A collapses on the second page along the diagonal), then is a universal additive et n Vassiliev invariant over A. Ren 2. Since is rationally injective, d is as well. Hence the Conjecture is true over Q. l Io T 3. Boavida-Weiss and BCKS group structures on agree.
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