By Andrew H Wallace

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**Example text**

Jones in [451. 12) was isolated by J. Wukai in [70]. 5. Framed manifolds It is not our business here to discuss the bordism theory of 2Z/2-framed manifolds, but simply to put the homotopytheoretic concepts of the last paragraph into a geometric setting. All manifolds will be smooth. For any closed manifold X (understood to be compact), with tangent bundle TX, the Pontrjagin-Thom construction defines what is variously called the index, direct image or Umkehr homomorphism: w*(X; -TX) -) w*(Point).

J,(X)) where t E w0'0(point) is defined as the class of (-1)+:(R ®L)+ 28 i*(t) = 1, p(t) = -1. --) (Y2(BL)+. The w 2-coefficient groups will be written: w i j : _ They are related to the non-equivariant groups 1'-j(point). wi in the following lemma. Let be w_1 'O be the Euler class of L. that is, the element represented by the inclusion 0+ -+ L+ (of the zero subspace in L) . 1*(b) = 0, p(b) = 1. i*: w of the trivial subgroup in 7Z/2. "i i is multiplication by (-1)i-j 1+ It is the 2Z/2-stable cohomotopy exact sequence of the pair (D(L),S(L)), with coefficients.

M, n E ]N (The limit is taken over the inclusions. copies of L; Ln means the sum of n in some contexts it is more natural to write nL. -J(X21/2). ) A product can be defined in w**(X) such that the maps i" and y are ring homomorphisms. -j,(X)) where t E w0'0(point) is defined as the class of (-1)+:(R ®L)+ 28 i*(t) = 1, p(t) = -1. --) (Y2(BL)+. The w 2-coefficient groups will be written: w i j : _ They are related to the non-equivariant groups 1'-j(point). wi in the following lemma. Let be w_1 'O be the Euler class of L.

### An introduction to algebraic topology by Andrew H Wallace

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