De Rham invariant




In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of Z/2{displaystyle mathbf {Z} /2}mathbf{Z}/2 – either 0 or 1. It can be thought of as the simply-connected symmetric L-group L4k+1,{displaystyle L^{4k+1},}{displaystyle L^{4k+1},} and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, L4k≅L4k{displaystyle L^{4k}cong L_{4k}}{displaystyle L^{4k}cong L_{4k}}), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant L4k+2.{displaystyle L_{4k+2}.}{displaystyle L_{4k+2}.}


It is named for Swiss mathematician Georges de Rham, and used in surgery theory.[1][2]



Definition


The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:[3]



  • the rank of the 2-torsion in H2k(M),{displaystyle H_{2k}(M),}{displaystyle H_{2k}(M),} as an integer mod 2;

  • the Stiefel–Whitney number w2w4k−1{displaystyle w_{2}w_{4k-1}}{displaystyle w_{2}w_{4k-1}};

  • the (squared) Wu number, v2kSq1v2k,{displaystyle v_{2k}Sq^{1}v_{2k},}{displaystyle v_{2k}Sq^{1}v_{2k},} where v2k∈H2k(M;Z2){displaystyle v_{2k}in H^{2k}(M;Z_{2})}{displaystyle v_{2k}in H^{2k}(M;Z_{2})} is the Wu class of the normal bundle of M{displaystyle M}M and Sq1{displaystyle Sq^{1}}{displaystyle Sq^{1}} is the Steenrod square ; formally, as with all characteristic numbers, this is evaluated on the fundamental class: (v2kSq1v2k,[M]){displaystyle (v_{2k}Sq^{1}v_{2k},[M])}{displaystyle (v_{2k}Sq^{1}v_{2k},[M])};

  • in terms of a semicharacteristic.



References





  1. ^ Morgan, John W; Sullivan, Dennis P. (1974), "The transversality characteristic class and linking cycles in surgery theory", Annals of Mathematics, 2, 99: 463–544, doi:10.2307/1971060, MR 0350748.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ John W. Morgan, A product formula for surgery obstructions, 1978


  3. ^ (Lusztig, Milnor & Peterson 1969)



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  • Lusztig, George; Milnor, John; Peterson, Franklin P. (1969), "Semi-characteristics and cobordism", Topology, 8: 357–360, doi:10.1016/0040-9383(69)90021-4, MR 0246308

  • Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980









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