# Wall invariant

An element of the Wall group, representing the obstruction to the surgery of a bordism to a simple homotopy equivalence.

Let $X$ be a finite Poincaré complex, $\nu$ a fibre bundle over $X$ and $x = [( M, \phi , F )] \in \Omega ( X, \nu )$ a bordism class, where $m$ is the formal dimension of $X$ and $\phi : M \rightarrow X$ has degree 1. This mapping can always be represented by an $[ m/2]$- connected mapping using a finite sequence of surgeries. Let $\Lambda = Z [ \pi _ {1} ( X)]$ be a group ring and let $\overline{ {}}\;$ be the involution on $\Lambda$ given by the formula $\overline{ {\sum _ {g} n ( g) g }}\; = \sum w ( g) n ( g) g ^ {-} 1$, where $w: \pi _ {1} ( X) \rightarrow \{ 1, - 1 \}$ is defined by the first Stiefel–Whitney class of $\nu$. Put

$$K ^ {*} ( M) = \ \mathop{\rm coker} ( \phi ^ {*} : H ^ {*} ( X) \rightarrow H ^ {*} ( M)),$$

$$K _ {*} ( M) = \mathop{\rm ker} ( \phi _ {*} : H _ {*} ( M) \rightarrow H _ {*} ( X))$$

(coefficients in $\Lambda$). The involution is an anti-isomorphism and there is defined the Wall group $U _ {n} ( \Lambda ) = L _ {n} ( \pi _ {1} ( X), w)$.

Suppose now that $m = 2k \geq 4$. Then in the stable free $\Lambda$- module $G = K _ {k} ( M) = \pi _ {k + 1 } ( \phi )$ one can choose a basis, and Poincaré duality induces a simple isomorphism $\lambda : G \rightarrow G ^ {*} = K ^ {k} ( M)$, where $( G, \lambda )$ is a $(- 1) ^ {k}$- form. One therefore obtains the class $\Theta _ {2k} ( x) = [( G, \lambda )] \in L _ {2k} ( \pi _ {1} ( X), w)$.

Suppose next that $m = 2k + 1 \geq 5$. One can choose generators in $\pi _ {k + 1 } ( \phi ) = K _ {k} ( M; \Lambda )$ so that they represent the imbeddings $f _ {i} : S ^ {k} \times D ^ {k + 1 } \rightarrow M$, with non-intersecting images, and these images are connected by paths with a base point. Put $U = \cup _ {i} \mathop{\rm Im} f _ {i}$, $M _ {0} = M \setminus \mathop{\rm Int} U$. Since $\phi \circ f _ {i} \sim 0$, one may replace $\phi$ by a homotopy and suppose that $\phi ( u) = *$. Because $X$ is a Poincaré complex, one can replace $X$ by a complex with a unique $m$- cell, i.e. one has a Poincaré pair $( X _ {0} , S ^ {m + 1 } )$ and $X = X _ {0} \cup e ^ {m}$. By the choice of a suitable cellular approximation one obtains a mapping for the Poincaré triad of degree 1: $\phi : ( M; M _ {0} , U) \rightarrow ( X; X _ {0} , e ^ {m} )$. Consequently one has the diagram of exact sequences:



Moreover, one has a non-degenerate pairing $\lambda : K _ {k} ( \partial U) \times K _ {k} ( \partial U) \rightarrow \Lambda$, where $H = ( K _ {k} ( \partial U), \lambda )$ is a quadratic $(- 1) ^ {k}$- form while $K _ {k + 1 } ( U, \partial U)$ and $K _ {k + 1 } ( M _ {0} , \partial U)$ define its Lagrange planes $L$ and $P$. Then $\Theta _ {2k + 1 } ( x) = [( H; L, P)] \in U _ {2k + 1 } ( \Lambda ) = L _ {2k + 1 } ( \pi _ {1} ( x), w)$.

The elements $\Theta _ {m} ( x) \in L _ {m} ( \pi _ {1} ( x), w)$ defined above are called the Wall invariants. An important property is the independence of $\Theta ( x)$ on the choices in the construction and the equivalence of the equation $\Theta ( x) = 0$ to the representability of the class as a simple homotopy equivalence, cf. [1].

#### References

 [1] C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970) [2] A.A. Ranicki, "The algebraic theory of surgery I" Proc. London Math. Soc. , 40 : 1 (1980) pp. 87–192 [3] S.P. Novikov, "Algebraic construction and properties of Hermitian analogs of -theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes I" Math. USSR Izv. , 4 : 2 (1970) pp. 257–292 Izv. Akad. Nauk SSSR Ser. Mat. , 34 : 2 (1970) pp. 253–288
How to Cite This Entry:
Wall invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wall_invariant&oldid=49168
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article