Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

Скачать книгу

Π₂} is called cancelable in the diagram Δ.

      Such pairs of cells are called cancelable because if a diagram Δ over a group G on a surface S has a pair of cancelable -cells, there exists a diagram Δ′ over the group G with two fewer -cells on the same surface S. Moreover, if the surface S has boundary, then the cancellation of cells of the diagram Δ leaves the labels of its contours unchanged.

      Given a diagram Δ and performing cell cancellation we get a diagram Δ′ with no cancelable pairs of cells. Such diagrams are called reduced. Since this process of reduction does not change the boundary label of a diagram, we obtain the following enhancements of Lemma 1.1 and Lemma 1.2:

Theorem 1.1. Let W be an arbitrary non-empty word in the alphabet ¹. Then W =1 in a group G given by its presentation (1.1) if and only if there exists a reduced disc diagram over the presentation (1.1) such that the label of its contour graphically equals W.

      Theorem 1.2. Let V, W be two arbitrary non-empty words in the alphabet ¹. Then they are conjugate in a group G given by its presentation (1.1) if and only if there exists a reduced annular diagram over the presentation (1.1) such that it has to contours p and q with the labels φ(p) ≡ V and φ(q) = W⁻¹.

1.1.4Unoriented diagrams

      In the present section we introduce a notion of unoriented diagrams — a slight modification of van Kampen diagrams which is useful in the study of a certain class of groups.

      Consider a diagram Δ over a group G with a presentation (1.1). With the alphabet we associate an alphabet which is in a bijection with the alphabet and is an image of the natural projection π : ¹ → defined as

      for all a ∈ with being the corresponding element of .

      Now we take the diagram Δ and “forget” the orientation of its edges. The resulting 1-complex will be called an unoriented diagram over the group G and denoted by . All definitions for diagrams (such as disc and annular diagrams, cells, contours, labels, etc.) are repeated verbatim for unoriented diagrams.

      

      Since each edge of the diagram Δ was decorated with a letter from the alphabet and we obtained from Δ just by forgetting the orientation of the edges, each edge of the diagram is decorated with an element of the alphabet as well. Therefore, walking around the boundary of a cell in a chosen direction we obtain a sequence of letters but, unlike the oriented case, we do not have an orientation of the edges to determine the sign of each appearing letter. Therefore we shall say that the label φ(∂Π) of the contour of a cell Π of the diagram is a cyclic word in the alphabet .

      Given a word in the alphabet we can produce 2ⁿ words in the alphabet ¹ of the form with εi ∈ {+1, −1}. We shall call each of those words a resolution of the word .

      Unoriented diagrams are very useful when describing group presentations such that the relation a² = 1 holds for all generators of the group, in other words, groups with the presentation

      In fact, the following analog of the van Kampen lemma holds:

      Lemma 1.4. Let W be an arbitrary non-empty word in the alphabet ¹. Then W = 1 in a group G given by its presentation (1.2) if and only if there exists a reduced unoriented disc diagram over the presentation (1.2) such that there is a resolution of the label of its contour which graphically equals W.

      Proof. First let W be a non-empty word in the alphabet ¹ such that W = 1 in the group G. Let us show that there exists an unoriented diagram with the corresponding label of its contour.

      Since W = 1, due to the strong van Kampen lemma (Theorem 1.1) there exists a reduced disc diagram over the presentation (1.2) such that the label of its contour graphically equals W. Denote this diagram by Δ. Now transform every edge ei of this diagram into a bigon with the label φ(ei)². Note that the result of this transformation is still a disc diagram. Indeed, we replaced every edge with a -cell (since φ(ei)² = 1 in the group G) and reversed the orientation of some edges in the boundary contours of the cells of the diagram Δ but they remain -cells due to the relations a² = 1.

      Now we “collapse” those 0-cells: replace every bigonal cell with the label of the form φ(e)² with an unoriented edge. Thus we obtain an unoriented diagram and the label

Скачать книгу