Algebraic and Geometric intersection numbers for free groups

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.Comment: 7 page

(Achiral) Lefschetz fibration embeddings of 44-manifolds

In this paper, we prove Lefschetz fibration embeddings of achiral as well as simplified broken (achiral) Lefschetz fibrations of compact, connected, orientable $4$-manifolds over $D^2$ into the trivial Lefschetz fibration of $\mathbb CP^2\times D^2$ over $D^2$. These results can be easily extended to achiral as well as simplified broken (achiral) Lefschetz fibrations over $\mathbb CP^1.$ From this, it follows that every closed, connected, orientable $4$-manifold admits a smooth (simplified broken) Lefschetz fibration embedding in $\mathbb CP^2\times \mathbb CP^1.$ We provide a huge collection of bordered Lefschetz fibration which admit bordered Lefschetz fibration embeddings into a trivial Lefschetz fibration $\tilde\pi:D^4\times D^2\to D^2.$ We also show that every closed, connected, orientable $4$-manifold $X$ admits a smooth embedding into $S^4\times S^2$ as well as into $S^4\tilde\times S^2$. From this, we get another proof of a theorem of Hirsch which states that every closed, connected, orientable $4$-manifold smoothly embeds in $\mathbb R^7.$ We also discuss Lefschetz fibration embedding of non-orientable $4$-manifolds $X$, where $X$ does not admit $3$- and $4$-handles in the handle decomposition, into the trivial Lefschetz fibration of $\mathbb CP^2\times D^2$ over $D^2$.Comment: 38 Pages; 21 Figures; V2- We added results on Lefschetz fibration embedding of non-orientable manifolds. V3- We added result on smooth embedding of $4$-manifolds in $S^2\times S^4$ as well as in $S^2\tilde\times S^4$. V4- We added corollary which gives an embedding of $4$-manifolds in $\mathbb R^7$. V5, V6- Typos and some minor technical errors are corrected. V7- Theorems 29, 31, 32 are adde

Geosphere laminations in free groups

We construct for free groups, which are codimension one analogues of geodesic laminations on surfaces. Other analogues that have been constructed by several authors are dimension-one instead of codimension-one. Our main result is that the space of such laminations is compact. This in turn is based on the result that crossing, in the sense of Scott-Swarup, is an open condition. Our construction is based on Hatcher's normal form for spheres in the model manifold

Splittings of free groups, normal forms and partitions of ends

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # (k) S (2) x S (1). These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in pi (2)(M) can be represented by an embedded sphere

Disseminated M. bovis Infection and Vertebral Osteomyelitis following Immunotherapy for Bladder Cancer

The use of BCG in immunotherapy for bladder cancer has been in practice for over 40 years. However, uncommon, serious complications can occur with the therapy. Here, we present a case of vertebral osteomyelitis secondary to dissemination of BCG following immunotherapy, an exceedingly rare presentation of an already rare complication