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Fulminant intracranial hypertension as a result of otological surgery: case report and discussion of management
- A Morris, J N Higgins, P R Axon
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- Journal:
- The Journal of Laryngology & Otology / Volume 135 / Issue 6 / June 2021
- Published online by Cambridge University Press:
- 18 May 2021, pp. 551-554
- Print publication:
- June 2021
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Background
Dural venous sinus injury is a rare complication of otological surgery that can lead to life-threatening sequelae, the management of which is complex and poorly described.
Case reportThis paper describes the case of a 40-year-old female who underwent routine right myringoplasty complicated by sigmoid sinus laceration. The patient subsequently developed right-sided lateral sinus thrombosis leading to fulminant intracranial hypertension. The patient underwent successful emergency management by surgical reconstruction of the sigmoid sinus, followed by endovascular thrombolysis, catheter balloon angioplasty and endovascular stenting.
ConclusionTorrential haemorrhage following otological procedures is uncommon and rarely requires packing of a bleeding venous sinus. This case highlights that injury to a highly dominant venous sinus can lead to venous outflow obstruction and life-threatening intracranial hypertension. To our knowledge, the development of this complication following otological surgery and its management has not been reported previously.
A global threats overview for Numeniini populations: synthesising expert knowledge for a group of declining migratory birds
- JAMES W. PEARCE-HIGGINS, DANIEL J. BROWN, DAVID J. T. DOUGLAS, JOSÉ A. ALVES, MARIAGRAZIA BELLIO, PIERRICK BOCHER, GRAEME M. BUCHANAN, ROB P. CLAY, JESSE CONKLIN, NICOLA CROCKFORD, PETER DANN, JAANUS ELTS, CHRISTIAN FRIIS, RICHARD A. FULLER, JENNIFER A. GILL, KEN GOSBELL, JAMES A. JOHNSON, ROCIO MARQUEZ-FERRANDO, JOSE A. MASERO, DAVID S. MELVILLE, SPIKE MILLINGTON, CLIVE MINTON, TAEJ MUNDKUR, ERICA NOL, HANNES PEHLAK, THEUNIS PIERSMA, FRÉDÉRIC ROBIN, DANNY I. ROGERS, DANIEL R. RUTHRAUFF, NATHAN R. SENNER, JUNID N. SHAH, ROB D. SHELDON, SERGEJ A. SOLOVIEV, PAVEL S. TOMKOVICH, YVONNE I. VERKUIL
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- Journal:
- Bird Conservation International / Volume 27 / Issue 1 / March 2017
- Published online by Cambridge University Press:
- 01 March 2017, pp. 6-34
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The Numeniini is a tribe of 13 wader species (Scolopacidae, Charadriiformes) of which seven are Near Threatened or globally threatened, including two Critically Endangered. To help inform conservation management and policy responses, we present the results of an expert assessment of the threats that members of this taxonomic group face across migratory flyways. Most threats are increasing in intensity, particularly in non-breeding areas, where habitat loss resulting from residential and commercial development, aquaculture, mining, transport, disturbance, problematic invasive species, pollution and climate change were regarded as having the greatest detrimental impact. Fewer threats (mining, disturbance, problematic native species and climate change) were identified as widely affecting breeding areas. Numeniini populations face the greatest number of non-breeding threats in the East Asian-Australasian Flyway, especially those associated with coastal reclamation; related threats were also identified across the Central and Atlantic Americas, and East Atlantic flyways. Threats on the breeding grounds were greatest in Central and Atlantic Americas, East Atlantic and West Asian flyways. Three priority actions were associated with monitoring and research: to monitor breeding population trends (which for species breeding in remote areas may best be achieved through surveys at key non-breeding sites), to deploy tracking technologies to identify migratory connectivity, and to monitor land-cover change across breeding and non-breeding areas. Two priority actions were focused on conservation and policy responses: to identify and effectively protect key non-breeding sites across all flyways (particularly in the East Asian- Australasian Flyway), and to implement successful conservation interventions at a sufficient scale across human-dominated landscapes for species’ recovery to be achieved. If implemented urgently, these measures in combination have the potential to alter the current population declines of many Numeniini species and provide a template for the conservation of other groups of threatened species.
Contributors
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O’Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
- Edited by Robert Audi, University of Notre Dame, Indiana
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- The Cambridge Dictionary of Philosophy
- Published online:
- 05 August 2015
- Print publication:
- 27 April 2015, pp ix-xxx
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Intellectual abilities in tuberous sclerosis complex: risk factors and correlates from the Tuberous Sclerosis 2000 Study
- P. F. Bolton, M. Clifford, C. Tye, C. Maclean, A. Humphrey, K. le Maréchal, J. N. P. Higgins, B. G. R. Neville, F. Rijsdjik, The Tuberous Sclerosis 2000 Study Group, J. R. W. Yates
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- Psychological Medicine / Volume 45 / Issue 11 / August 2015
- Published online by Cambridge University Press:
- 01 April 2015, pp. 2321-2331
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Background
Tuberous sclerosis complex (TSC) is associated with intellectual disability, but the risk pathways are poorly understood.
MethodThe Tuberous Sclerosis 2000 Study is a prospective longitudinal study of the natural history of TSC. One hundred and twenty-five UK children age 0–16 years with TSC and born between January 2001 and December 2006 were studied. Intelligence was assessed using standardized measures at ≥2 years of age. The age of onset of epilepsy, the type of seizure disorder, the frequency and duration of seizures, as well as the response to treatment was assessed at interview and by review of medical records. The severity of epilepsy in the early years was estimated using the E-Chess score. Genetic studies identified the mutations and the number of cortical tubers was determined from brain scans.
ResultsTSC2 mutations were associated with significantly higher cortical tuber count than TSC1 mutations. The extent of brain involvement, as indexed by cortical tuber count, was associated with an earlier age of onset and severity of epilepsy. In turn, the severity of epilepsy was strongly associated with the degree of intellectual impairment. Structural equation modelling supported a causal pathway from genetic abnormality to cortical tuber count to epilepsy severity to intellectual outcome. Infantile spasms and status epilepticus were important contributors to seizure severity.
ConclusionsThe findings support the proposition that severe, early onset epilepsy may impair intellectual development in TSC and highlight the potential importance of early, prompt and effective treatment or prevention of epilepsy in tuberous sclerosis.
RANK PROPERTIES OF ENDOMORPHISMS OF INFINITE PARTIALLY ORDERED SETS
- P. M. HIGGINS, J. D. MITCHELL, M. MORAYNE, N. RUšKUC
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- Journal:
- Bulletin of the London Mathematical Society / Volume 38 / Issue 2 / April 2006
- Published online by Cambridge University Press:
- 16 March 2006, pp. 177-191
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- April 2006
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The relative rank $\mbox{\rm rank}(S:U)$ of a subsemigroup $U$ of a semigroup $S$ is the minimum size of a set $V\subseteq S$ such that $U$ together with $V$ generates the whole of $S$. As a consequence of a result of Sierpiński, it follows that for $U\leq \mathcal{T}_{X}$, the monoid of all self-maps of an infinite set $X$, $\mbox{\rm rank}(\mathcal{T}_{X}:U)$ is either 0, 1 or 2, or uncountable. In this paper, the relative ranks $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})$ are considered, where $X$ is a countably infinite partially ordered set and $\mathcal{O}_{X}$ is the endomorphism monoid of $X$. We show that $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})\leq 2$ if and only if either: there exists at least one element in $X$ which is greater than, or less than, an infinite number of elements of $X$; or $X$ has $|X|$ connected components. Four examples are given of posets where the minimum number of members of $\mathcal{T}_{X}$ that need to be adjoined to $\mathcal{O}_{X}$ to form a generating set is, respectively, 0, 1, 2 and uncountable.
Two stage management of petrositis with associated mycotic aneurysm of the intrapetrous carotid artery
- G J Crossland, R De, J N P Higgins, P R Axon
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- Journal:
- The Journal of Laryngology & Otology / Volume 119 / Issue 6 / June 2005
- Published online by Cambridge University Press:
- 08 March 2006, pp. 479-482
- Print publication:
- June 2005
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Mycotic aneurysm of the petrous temporal bone is extremely rare, with only 12 cases previously reported. We review the literature to date and present a case of petrositis complicated by a mycotic aneurysm of the internal carotid artery, which was managed by endovascular balloon occlusion and subsequent total petrosectomy. We can find no other case in the literature where balloon occlusion has been used to treat a mycotic aneurysm of the intrapetrous carotid artery prior to total petrosectomy to treat petrositis.
COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS
- P. M. Higgins, J. M. Howie, J. D. Mitchell, N. Ruškuc
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 46 / Issue 3 / October 2003
- Published online by Cambridge University Press:
- 10 December 2003, pp. 531-544
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The relative rank $\rank(S:A)$ of a subset $A$ of a semigroup $S$ is the minimum cardinality of a set $B$ such that $\langle A\cup B\rangle=S$. It follows from a result of Sierpiński that, if $X$ is infinite, the relative rank of a subset of the full transformation semigroup $\mathcal{T}_{X}$ is either uncountable or at most $2$. A similar result holds for the semigroup $\mathcal{B}_{X}$ of binary relations on $X$.
A subset $S$ of $\mathcal{T}_{\mathbb{N}}$ is dominated (by $U$) if there exists a countable subset $U$ of $\mathcal{T}_{\mathbb{N}}$ with the property that for each $\sigma$ in $S$ there exists $\mu$ in $U$ such that $i\sigma\le i\mu$ for all $i$ in $\mathbb{N}$. It is shown that every dominated subset of $\mathcal{T}_{\mathbb{N}}$ is of uncountable relative rank. As a consequence, the monoid of all contractions in $\mathcal{T}_{\mathbb{N}}$ (mappings $\alpha$ with the property that $|i\alpha-j\alpha|\le|i-j|$ for all $i$ and $j$) is of uncountable relative rank.
It is shown (among other results) that $\rank(\mathcal{B}_{X}:\mathcal{T}_{X})=1$ and that $\rank(\mathcal{B}_{X}:\mathcal{I}_{X})=1$ (where $\mathcal{I}_{X}$ is the symmetric inverse semigroup on $X$). By contrast, if $\mathcal{S}_{X}$ is the symmetric group, $\rank(\mathcal{B}_{X}:\mathcal{S}_{X})=2$.
AMS 2000 Mathematics subject classification: Primary 20M20
GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS
- P. M. HIGGINS, J. D. MITCHELL, N. RUšKUC
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- Journal:
- Glasgow Mathematical Journal / Volume 45 / Issue 3 / September 2003
- Published online by Cambridge University Press:
- 10 September 2003, pp. 557-566
- Print publication:
- September 2003
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For a linearly ordered set $X$ we consider the relative rank of the semigroup of all order preserving mappings $\mathcal{O}_{X}$ on $X$ modulo the full transformation semigroup $\mathcal{T}_{X}$. In other words, we ask what is the smallest cardinality of a set $A$ of mappings such that $\genset{\mathcal{O}_{X}\cup A}=\mathcal{T}_{X}$. When $X$ is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when $X=\mathbb{R}$ (the set of real numbers) it is uncountable.
The non-abelian tensor product of groups and related constructions
- N. D. Gilbert, P. J. Higgins
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- Journal:
- Glasgow Mathematical Journal / Volume 31 / Issue 1 / January 1989
- Published online by Cambridge University Press:
- 18 May 2009, pp. 17-29
- Print publication:
- January 1989
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The tensor product of two arbitrary groups acting on each other was introduced by R. Brown and J.-L. Loday in [5, 6]. It arose from consideration of the pushout of crossed squares in connection with applications of a van Kampen theorem for crossed squares. Special cases of the product had previously been studied by A. S.-T. Lue [10] and R. K. Dennis [7]. The tensor product of crossed complexes was introduced by R. Brown and the second author [3] in connection with the fundamental crossed complex π(X) of a filtered space X, which also satisfies a van Kampen theorem. This tensor product provides an algebraic description of the crossed complex π(X ⊗ Y) and gives a symmetric monoidal closed structure to the category of crossed complexes (over groupoids). Both constructions involve non-abelian bilinearity conditions which are versions of standard identities between group commutators. Since any group can be viewed as a crossed complex of rank 1, a close relationship might be expected between the two products. One purpose of this paper is to display the direct connections that exist between them and to clarify their differences.