Working with Intersection Forms in Homology. Computation. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:12:05Z http://mathoverflow.net/feeds/question/22238 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22238/working-with-intersection-forms-in-homology-computation Working with Intersection Forms in Homology. Computation. HErb 2010-04-22T19:01:58Z 2010-05-07T04:45:18Z <p>Hi, everyone:</p> <p>I am trying to work with the intersection form in 4-manifolds. Specifically,</p> <p>I am working with \$CP^2\$ (complex projective 2-space.), whose form is given by \$(1)\$.</p> <p>Now, I know how to compute an actual numerical value when we work with the form in cohomology: we cup-product two cochains a,b , and then evaluate \$a \cup b\$ on the fundamental class.</p> <p>But when we work in homology (using Poincare Duality) , I am not too clear on how we actually get a number by starting with a matrix (we always have representative surfaces for 2-homology in a 4-manifold.). What do we evaluate this matrix in.?</p> <p>Thanks. </p> http://mathoverflow.net/questions/22238/working-with-intersection-forms-in-homology-computation/22282#22282 Answer by Herb for Working with Intersection Forms in Homology. Computation. Herb 2010-04-23T00:35:51Z 2010-04-23T02:00:40Z <p>Thanks , David, both for the formatting and the ref. Unfortunately, I think my question may be much simpler than your refs: I have a matrix representation of a form in cohomology , which can be dualized (and I give this dualized form) to homology. This form/matrix is supposed to output an integer value; this value is the number of points of intersection of two submanifolds of a 4-manifold, with the sign having to see with the orientation of the two submanifolds.</p> <p>I am just not clear on how I can get this integer value from the matrix, i.e., how I can get the intersection number using this form; I know I need to evaluate this matrix on some 2x1 vector, I just have no idea of what this vector would be.</p> <p>Herb.</p> http://mathoverflow.net/questions/22238/working-with-intersection-forms-in-homology-computation/23821#23821 Answer by S. Carnahan for Working with Intersection Forms in Homology. Computation. S. Carnahan 2010-05-07T04:45:18Z 2010-05-07T04:45:18Z <p>If you have 2 surfaces in a 4-manifold, they represent two elements (say, \$a\$ and \$b\$) in the degree 2 homology. If you have a matrix representation \$M\$ of the intersection form, this means you have already chosen a basis of degree 2 homology, and you can express \$a\$ and \$b\$ as column vectors with respect to this basis. You get an integer by taking \$a^TMb\$, where \$a^T\$ denotes the transpose of \$a\$.</p>