Let A: M → N be a morphism in X −Cobn. We call A a cylinder if A ∼= M×I
as topological manifolds. And A is an X-cylinder of M or simply an X-cylinder ifA is
X-homeomorphic to (M ×I, fM, α), with α given as identity on one-end and boundary
map of A at the other end. Clearly, an X-cylinder is a cylinder, but a cylinder is not necessarily anX-cylinder. For example, consider the handle∅ →S1tS1. The concept of cylinders andX-cylinders have two applications, or rather two directions to go about. One is considering the mapping class group of Mcontaining all the equivalent classes of the
X-cylinders of M. And the other direction is to enrich the collection of circles in X with a structure of Turaev crossed system usingX-cylinders of circles. Let us denote the set of
representatives of equivalence classes of 1-morphisms betweenMand K as Hom (M,K), then AΨ ∈ Hom (M,K) and proposition (3.3.1) implies that AΨ and AΦ represent the same element in Hom (M,K) when Ψ and Φ are isotopic. Note thatAψ ∈Hom (M,M), when ψis a self X-homeomorphism of M.
Let HomeoX(M) be the group of self X-homeomorphisms of Mwith compact open
topology. Let HomeoX0(M) be the subset of HomeoX(M) consisting of allX-homeomorphisms
isotopic to the identity morphism on M. It is easy to verify that HomeoX0 (M) is in fact
a normal open subgroup in HomeoX(M). The factor group
MCGX(M) = HomeoX(M)/HomeoX0 (M)
is the mapping class group of M. The elements of this group are the isotopy classes of selfX-homeomorphisms ofM. Note thatX-cylinders define a semigroup homomorphism from MCGX(M) to Hom (M,M) given by [ψ]7→ Aψ and the composition is given by :
[ψ◦φ]7→ Aψ◦φ=Aψ◦ Aφ.
Let us now work in the other direction to provide the circles with a structure of a Turaev G-crossed system.
Using these morphisms, we conclude the section with the following result:
Theorem 3.4.5 Let Gbe a group and X a K(G; 1) space. Then inX −Cob1, cylinders
define a TuraevG-crossed system on circles.
Proof: We have discussed in the Proposition 3.4.4 that circles form a Frobenius G-
graded system inX −Cob1. The cylinder C−−(g,1) defines a form φg :Ag⊗Ag−1 −→I
on circles. The information of G-action on circles is carried by the cylinder C−+(g,h−1),
ı.e. φg :Ag⊗Ag−1 C−−(g,1) −−−−−−→I ϕg,h :Ag C−+(g,h −1) −−−−−−−→Ahgh−1
We show the axioms of a Turaev crossed system in the following seven easy steps: (i) ϕf,gh = ϕhf h−1,g ◦ ϕf,h is depicted in Figure 3.11. Observe that the gluing of
Gluing
Figure 3.11: The non-degenerate symmetric form on circles.
using the composition of the cylinders inX −Cob1.
(ii) The way we have defined the action, we have
ϕg,1 :Ag −→Ag
is an identity morphism onAg. This is so becauseAg = (S−1, g) and theX-morphism
ϕg,1 is the X-cylinder C−+(g,1) mappingS1 toS1 as an identity map.
(iii) We show that the action is an algebra morphism. GluingC−+(f g, h−1) toP−−+(f, g,1,1)
along (S−1, f g) = (C+0, f g), we obtain the X-morphismP−−+(f, g, h, h).
Similarly , gluingC−+(f, h−1)∪C−+(g, h−1) toP−−+(hf h−1f, hgh−1,1,1) along the
circles (S−1, hf h−1) = (L−, hf h−1) and (S−1, hgh−1) = (M−, hgh−1) respectively, we
again obtain the same X-morphismP−−+(f, g, h, h). Therefore,
ϕf g,h◦µf,g =µhf h−1,hgh−1 ϕf,h⊗ϕg,h
.
(iv) We argue that the action preserves form as follows. Assume that g = f−1. Now,
gluingC−+(f, h−1)∪C−+(g, h−1) to the cylinder C−−(hf h−1,1) along the circles
(S−1, hf h−1) = (L−, hf h−1) and,
(S−1, hgh−1) = (M−, hgh−1)
respectively, we obtain the X-morphismC−−(f,1). Thus,
(v) The Dehn twist along the circleS1×(1/2)⊂C−
+(f,1) yields anX-homeomorphism
between C−+(f,1) and C−+(f, f). Thus they are in the same equivalence class of
isomorphism in X −Cob1. Hence
ϕf,f =ϕf,1= id :Af −→Af.
(vi) Consider a self X-homeomorphismζ of the (2,X)-pantsP which is the identity on
N and which permutes (L, l) and (M, m). We chooseζso thatζ(nm) =nlandζ(nl) is an embedded arc leading fromntomand homotopic to the product of four arcs:
nl, ∂L, (nl)−1, nm. An easy computation shows that ζ is an X-homeomorphism
fromP−−+(f, g,1,1) :Af⊗Ag →Af g toP−−+(g, f,1, g−1) :Ag⊗Af →Af g. Thus
they are obtained from each other by the permutation of the two tensor factors. Thus, this shows the commutativity of the following diagram:
Af ⊗Ag µf,g // τ Af g Ag⊗Af φf⊗1 / /Af gf−1⊗Af µf gf−1,g O O
This topologically can be interpreted as the X-homeomorphism between the two cobordisms below: The arrow in the above figure implies that theX-morphism on
𝑓𝑔 𝑓𝑔 𝑔 𝑓 𝐶−+(𝑓, 1) 𝑃−−+(𝑓, 𝑔, 1,1) 𝑃𝑒𝑟𝑚𝑢𝑡𝑖𝑛𝑔 𝑡𝑒 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 𝑃−−+(𝑓𝑔𝑓−1, 𝑓, 1,1) 𝑓𝑔𝑓−1 𝑓 𝑓 𝑔 𝐶−+(𝑔, 𝑓−1) 𝑺𝒑𝒍𝒊𝒕𝒕𝒊𝒏𝒈 𝑓 𝑔
Figure 3.12: Topological interpretation of the above commutative diagram
left handside of the arrow is obtained from the composition ofX–morphisms on the right.
(vii) Trace condition. For any f, g ∈ G, let Af = (S−1, f) and Ag = (S−1, g) be 0-
following compositions of X-morphisms: Hence,
Gluing
Gluing
Figure 3.13: Morphisms for trace axiom
ρf =µf gf1g1,gf g−1(1⊗ϕg) :Ah⊗Af →Af
ρg = (ϕf−1)µf gf1g1,g :Ah⊗Ag →Ag.
The trace axiom requires that the above two morphisms inX −Cob1 have the same
trace. Thus if
TrX : Hom (Ah⊗X, X)→Hom (Ah, I)
is the trace morphism related to 0-morphismX inX −Cob1, then we need to show
that
TrAf(ρf) = TrAg(ρg)
where each side is a morphism inX −Cob1 fromAh toI given by figure (3.14). We
I
Figure 3.14: Trace Axiom
now construct a morphismH= (H, F, αH) inX −Cob1 fromAh toI such that H
will be the partial trace forρf as well asρg.
Consider S−1 and fix one of its point, say s. Consider the 2-torus S−1 ×S−1 with
the loops S1
− ×s and s×S1− for s ∈ S−1. Consider the punctured torus H =
(S−1 ×S1−)/ IntB with orientation induced from S−1 ×S−1. Let us provide the
boundary circle ∂H = ∂B with orientation opposite to the one induced from H. We choose a base point on ∂H and an arc r ⊂H joining this point to s×s∈H. We can assume thatr meets the loopsS−1 ×sands×S−1 only in its endpoints×s.
Consider a mapF :H →X=K(G,1) such thatF(r) =x∈X and the restrictions ofF toS1
−×s,s×S−1 representf,g∈G, respectively. (Note that the orientations
of S−1 ×s, s×S−1 are induced by the one of S−1.) Then the loop F|∂H represents
f gf−1g−1. Now, H= (H, F, αH) is 1-morphism from (∂H−, F|∂H) = (Ah, h) to I.
Thus we have a morphism, H:Ah→I.
NowHcan be obtained fromP−−+(f gf−1g−1, f,1, g) by gluing the boundary com-
ponents (M−, f) and (N+, f) along an X-homeomorphism. (Note that the circles M− and N+ give the loop S1 ×s ⊂ N). A standard argument in the theory of
TQFTs shows that the homomorphism H : Af gf−1g−1 → I is the partial trace of
the homomorphism
ρf =P−−+(f gf−1g−1, f,1, g) :Af gf−1g−1⊗Af →Af.
Similarly,His obtained from P−−+(f gf−1g−1, g, f−1, f−1) by gluing the boundary
components (M−, g) and (N+, g) along anX-homeomorphism. (Note that the circles M− andN+ give the loop s×S1∈N.) Thus,H:Af gf−1g−1,f,1,g→I is the partial
trace of the homomorphism
ρg=P−−+(f gf−1g−1, g, f−1, f−1) :Af gf−1g−1 ⊗Ag→Ag.
2
Suppose X is a K(G; 1) space. We can extend the above result to formulate the data given by any X-HQFT with values in a monoidal category C to define a Turaev
G-crossed system in C. Suppose an X-HQFT (Z, τ) with values in C be given. Then instead of working with circles and cylinders in X −Cob1, we rather can work with the
Observe first that a 1-dimensional connectedX-manifoldMis just a pointed oriented circle endowed with a map into X sending the base point into x. This is nothing but a loop in X with endpoints in x. If ZM is the object inC given by (Z, τ) corresponding to
M, then clearly it depends only on the class of the loop in π1(X) = G. Thus for each g∈G, we obtain an object Ag in C.
Recall from Section (3.4.3), where we set our notation as Ag for (S−1;g). Instead (or,
by abuse of notation), we have now set Ag as an object in C given by a X-HQFT (Z, τ) which corresponds to a 1-dimensional connected X-manifold Mwhich depends only on the class g of the loop inG. Thus we have a collection A ={Ag} of objects in C. The
X-HQFT sends X-cobordisms to morphisms inC. Then, as in equations (3.11) to (3.14), we have the morphisms
µf,g :Af ⊗Ag →Af g
η:IC →A1
:A1 →IC
φf :Af ⊗Af−1 →I
inC which equips the collectionA={Ag}with a Frobenius structure (Proposition 3.4.3). Further, the calculations done in Theorem (3.4.4) are exactly the same so as to endow the Frobenius systemA={Ag}with a Turaev structure. Thus theX-HQFT (Z, τ) defines a
Turaev G-crossed systemA ={Ag} inC. We can summarise this discussion in the form
of the following result:
Theorem 3.4.6 Suppose X is a K(G; 1) space. Then any (1+1)-dimensional X-HQFT with values in C defines a Turaev G-crossed system in C.