signature (topology)

Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group

:$H^\{2k\}(M,\backslash mathbf\{R\})$.

The basic identity for the cup product

:$\backslash alpha^p\; \backslash smile\; \backslash beta^q\; =\; (-1)^\{pq\}(\backslash beta^q\; \backslash smile\; \backslash alpha^p)$

shows that with p = q = 2k the product is symmetric. It takes values in

:$H^\{4k\}(M,\backslash mathbf\{R\})$.

If we assume also that M is compact, Poincaré duality identifies this with

:$H\_\{0\}(M,\backslash mathbf\{R\})$

which can be identified with $\backslash mathbf\{R\}$. Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H^2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself.{{cite book|last1=Hatcher|first1=Allen|title=Algebraic topology|date=2003|publisher=Cambridge Univ. Pr.|location=Cambridge|isbn=978-0521795401|page=250|edition=Repr.|url=https://pi.math.cornell.edu/~hatcher/AT/AT.pdf|accessdate=8 January 2017|language=en}} More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.

The signature $\backslash sigma(M)$ of M is by definition the signature of Q, that is, $\backslash sigma(M)\; =\; n\_+\; -\; n\_-$ where any diagonal matrix defining Q has $n\_+$ positive entries and $n\_-$ negative entries.{{cite book|last1=Milnor|first1=John|last2=Stasheff|first2=James|title=Characteristic classes|date=1962|publisher=Annals of Mathematics Studies 246|page=224|isbn=978-0691081229|language=en|citeseerx=10.1.1.448.869}} If M is not connected, its signature is defined to be the sum of the signatures of its connected components.

{{details|L-theory}}

If M has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in L-theory: the signature can be interpreted as the 4k-dimensional (simply connected) symmetric L-group $L^\{4k\},$ or as the 4k-dimensional quadratic L-group $L\_\{4k\},$ and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of $\backslash mathbf\{Z\}/2$) for framed manifolds of dimension 4k+2 (the quadratic L-group $L\_\{4k+2\}$), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group $L^\{4k+1\}$); the other dimensional L-groups vanish.

{{main|Kervaire invariant}}

When $d=4k+2=2(2k+1)$ is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant. The resulting invariant of a manifold is called the Kervaire invariant.

• Compact oriented manifolds M and N satisfy $\backslash sigma(M\; \backslash sqcup\; N)\; =\; \backslash sigma(M)\; +\; \backslash sigma(N)$ by definition, and satisfy $\backslash sigma(M\backslash times\; N)\; =\; \backslash sigma(M)\backslash sigma(N)$ by a Künneth formula.

• If M is an oriented boundary, then $\backslash sigma(M)=0$.

• René Thom (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its Pontryagin numbers.{{cite news|last1=Thom|first1=René|title=Quelques proprietes globales des varietes differentiables|publisher=Comm. Math. Helvetici 28 (1954), S. 17–86|url=https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf|accessdate=26 October 2019|language=fr}} For example, in four dimensions, it is given by $\backslash frac\{p\_1\}\{3\}$. Friedrich Hirzebruch (1954) found an explicit expression for this linear combination as the L genus of the manifold.

• William Browder (1962) proved that a simply connected compact polyhedron with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.

• Rokhlin's theorem says that the signature of a 4-dimensional simply connected manifold with a spin structure is divisible by 16.

• Hirzebruch signature theorem
• Genus of a multiplicative sequence
• Rokhlin's theorem

{{Reflist}}

{{DEFAULTSORT:Signature (Topology)}}

Category:Geometric topology

Category:Quadratic forms