The asphericity of random 2‐dimensional complexes. Issue 2 (12th April 2013)

• Record Type: Journal Article
• Authors: Costa, A.E.; Farber, M.
• Is Part Of: Random structures & algorithms. Volume 46:Issue 2(2015)
• Page Start: 261
• Page End: 273
• Publication Date: 2013-04-12
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20499
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We study random 2‐dimensional complexes in the Linial–Meshulam model and prove that for the probability parameter satisfying <disp-formula content-type="mathematics" id="rsa20499-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp15" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≪</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>46</mml:mn><mml:mo>/</mml:mo><mml:mn>47</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></disp-formula> a random 2‐complex <italic>Y</italic> contains several pairwise disjoint tetrahedra such that the 2‐complex <italic>Z</italic> obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex <italic>Z</italic> satisfies the Whitehead conjecture, i.e. any subcomplex <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp0m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Z</mml:mi><mml:mo>′</mml:mo><mml:mo>⊂</mml:mo><mml:mi>Z</mml:mi></mml:mrow></mml:math></alternatives></inline-formula> is aspherical. This implies that <italic>Y</italic> is homotopy equivalent to a wedge <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp38" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Z</mml:mi><mml:mo>∨</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>∨</mml:mo><mml:mo>…</mml:mo><mml:mo>∨</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> where <italic>Z</italic> is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions <disp-formula content-type="mathematics" id="rsa20499-disp-0002"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp2q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>ϵ</mml:mo></mml:mrow></mml:msup><mml:mo>, </mml:mo></mml:mrow></mml:math></alternatives></disp-formula> where <italic>c</italic> > 3 and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp5c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mo>ϵ</mml:mo><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>47</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, the complex <italic>Z</italic> is genuinely 2‐dimensional and in particular, it has sizable 2‐dimensional homology; it follows that in the indicated range of the probability parameter <italic>p</italic> the cohomological dimension of the fundamental group <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734hp4t" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20499:rsa20499-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mo>π</mml:mo><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015</p> </abstract>