Heutiges TOpologiethema (23.12.13)

Abstract

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Indtroduciton

Let $M$ be a connected oriented compact $3$-manifold $m:\partial ({\Sigma }_{g,1}\times [-1,1])\to M$ an orientiation preserving homeo. The zylinderset is actually easy to visualize. Just imagine fattening a genus $g$ surface. We get a volume with tubes inside, which escape through the one boundary component. The best visualisation is taking a block with $g$ handles on top. Through the bottom of the block we drill tunnels which go thrhoug the handles.

Definition. A pair $(M,m)$ is called a homology cylinder over ${\Sigma }_{g,1}$ if both $m_{\pm }$ induce the same map on the homology. Two pairs $(M,m),(M^{\prime },m^{\prime })$ are identified if there is homeo between them and both maps $m,m^{\prime }$ map from the previously definied zylinder.

The set of all homology cylinders will be denoted by $I{C}_{g,1}$.

We define a multiplication on two pairs by sticking them on top of each other. This makes $I{C}_{g,1}$ a monoid. The identity element is $({\Sigma }_{g,1},\times [-1,1],Id)$. In the above picture this however is not really a stacking but more a stack the bricks and loop the handles through the holes. Interestingly. if the hole are braidshaped (inside the bricks) then the multiplication corresponds to stacking braids. (However we dont need to have braids, we could have a weird tangle where the strings go back down and which can’t be detangled by composition).

We now define a function space $M_{g,1}$. Look at all automorphisms on ${\Sigma }_{g,1}$ which are the identity on the boundary component of ${\Sigma }_{g,1}$ We now define the subset $I_{g,1}$ which consists of all automorphismus which induce the identity of the first homology group auf ${\Sigma }_{g,1}$.

The map $I_{g,1}\to I{C}_{g,1}$ is an injective monoid homomorphism. (See paper for definition)

Motivation

The theory we’re developing here has different amount of progress in different dimensions.

• dim 2: There is a theory of the things we want to do in two dimensions developed by N-Sato-Suzuk 2022.
• dim 3: This uses clasper theory by Goussarov and Habiro. (cf. clasper in dim $\ge 4$ by Watanabe (2018))
• dim 4: Uses the Homology cobordism group

We want to study $I{C}_{g,1}$. How can we do this? One idea by Massuyeau-Meilhan 2013 is to use Reidemeister-Turaev torsion.

We now defined Reidemeister-Torsion. Noticible is thatthe speaker does not use a determinant in his definition. Instead everything is part of the abelianizations of $GL(F)$, which is isomorphic to $F^{\times }$ where the isomorphisms is given by the identity. (It does make sense to not use the determinant. After all the determinant is multiplicative for finite elements)

Non-commutative RT-version