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Write $f_x$ for the linear isometry specified at $x$. Choose a cofinal sequence $V_i$ from the indexing family $I$, and write the target part of the supplied subordinate flag for a family of linear isometries as $W_i$. After a cofinal enlargement it hasBoth flags exhaust their universe for spectra. Compactness of the parameter space ensures that a finite target stage can contain all images of each fixed source stage.

Let ${\xi }_i$ be the vector bundle on $X$ with fiberHere $\ominus$ is the orthogonal complement in the indicated ambient subspace. Its rank is $\dim W_i-\dim V_i$. Let $\mathrm{Th}({\xi }_i)$ be its Thom space, the disk bundle modulo its sphere bundle, so that every fiber is compactified and all its points at infinity become the single basepoint. For the zero bundle this convention gives $\mathrm{Th}(0_X)=X_{+}$. Define a target indexed prespectrum on the flag byThe smash product here is of based spaces, not yet the derived smash product of spectra.

To specify all structure maps, for $i\le j$ use the continuous fiberwise orthogonal isomorphismBoth sides identify with the complement of $f_x(V_i)$ in $W_j$. In particular this is the isometry obtained by orthogonally projecting onto $f_x(V_j\ominus V_i)$ and its complement; it is not an assertion that the displayed summands coincide separately. Pass to fiberwise one-point compactifications, identify the last summand with $V_j\ominus V_i$ using $f_x^{-1}$, and apply the source structure map. This definesMore explicitly, decompose $z+w\in W_j\ominus f_x(V_i)$, where $z\in W_j\ominus W_i$ and $w\in ({\xi }_i{)}_x$, as $w^{\prime }+f_x(v)$ with $w^{\prime }\in ({\xi }_j{)}_x$ and $v\in V_j\ominus V_i$. Send its pair with $e\in E(V_i)$ to the pair $(w^{\prime },{\sigma }_{V_i,V_j}(v,e))$ over $x$. The notation for finite vectors extends to the compactifications, sending infinities and basepoints to the basepoint. Orthogonal decomposition and associativity of the source structure maps prove the prespectrum identities.

Extend the indexed prespectrum from the cofinal flag to the target indexing family and apply spectrification $L$. The point-set definition isOn a map of topological spectra $u:E\to E^{\prime }$ it is induced at stage $i$ by $1_{\mathrm{Th}({\xi }_i)}\wedge u(V_i)$. Thus the prescribed isometry family, indexing family and flag have all been used. A common cofinal refinement identifies the resulting objects in the stable homotopy category; the point-set model retains the chosen flag. If $f$ is a constant standard inclusion, the complement bundles are trivial and this recovers the usual reindexed $X_{+}\wedge E$.

The associativity of twisted half-smash products follows from an actual isomorphism of the complement bundles. First work over compact parameter spaces, so that a common choice of flags can be made. Choose source, intermediate and target stages $V_i,W_i,Z_i$ satisfyingFor the two twisted half-smash products and their composite the relevant vector bundles have fibersPull the first two bundles back to $Y\times X$. Since $g_y$ is a linear isometry, the decomposition of $Z_i$ into $g_y(W_i)$ and its orthogonal complement gives the fiberwise vector bundle isomorphismIts inverse is given by the two orthogonal projections. The isomorphism varies continuously with $(y,x)$, and the ranks agree:The one-point compactification of an external orthogonal direct sum gives the smash product of its Thom spaces. Using $g_y$ to identify $g_y({\xi }_x)$ with ${\xi }_x$, the isomorphism therefore givesThe left side is a stage of $(Y\times X){⋉}_{h}E$ and the right side is the corresponding two-step prespectrum presentation of $Y{⋉}_g(X{⋉}_{f}E)$.

These stage comparisons commute with structure maps. Indeed, increasing a stage decomposes a suspension vector into its component in $g_y{f}_x(V_j\ominus V_i)$ and the two residual complement components. In either construction the same source component is pulled back to $V_j\ominus V_i$ and used in the same structure map of $E$; the residual components are left in the same final complement. Iterated orthogonal decomposition gives the same result whether the intermediate complement is separated first or last. They also commute with maps of $E$, since the comparison only rearranges the suspension coordinates.

For clarity about the intervening spectrification, maps out of $LP$ into a genuine indexed spectrum $T$ are precisely compatible prespectrum maps $P\to T$. At stage $i$, adjunction from the Thom coordinate identifies such a map with a continuous familywhere the last identification is the genuine indexed spectrum structure of $T$. Compatibility between stages is exactly compatibility of this family with the source structure maps. Thus maps out of the twisted construction are continuous families of maps over the specified linear isometries. A family of such maps over $g$ out of $X{⋉}_{f}E$ is, by applying this universal property a second time, exactly a family of maps out of $E$ over $g_y{f}_x$. This is the same orthogonal-coordinate correspondence just constructed; it respects the identifications imposed by each spectrification. Consequently it descends from the prespectrum presentations to the associated spectra. Common cofinal refinements remove any incompatibility between the independently chosen flags.

The printed hypothesis does not require $Y$ to be compact. For an arbitrary compactly generated parameter space $Y$, use the same construction on compact test spaces mapping into $Y$. Each such family admits the common flags above, and the comparisons agree under restriction because their formula is the canonical orthogonal projection. They therefore assemble into the continuous global comparison in the compactly generated point-set construction. Equivalently, the preceding universal-property correspondence is valid for arbitrary $Y$ and does not impose a uniform finite target stage on all of it. Passing to cellular models for the derived construction gives the requested natural isomorphism

The exactness axiom of a generalized cohomology theory gives a natural long exact sequence for each CW pair:For $(X,X)$, the restriction $h^n(X)\to h^n(X)$ is the identity. Its preceding relative-to-absolute map has zero image. The previous connecting map is zero too, since it follows the surjective identity in degree $n-1$. Exactness at the relative group therefore makes its map both injective and zero, givingin particular $h^n(x_0,x_0)=0$. No dimension axiom is being used; a generalized theory can have nonzero $h^n(x_0)$ in many degrees.

The wedge-product notation for pairs here means the smash product of CW pairs,Its quotient is the based smash product $(X/A)\wedge (Y/B)$. A multiplicative theory has an external productIf the two spaces are the same, pullback by the diagonal gives the relative cup product in $h^{k+m}(X,A\cup B)$. This is different from the wedge sum of two based spaces.

First read the covering subcomplexes as pointed subcomplexes, so $x_0\in A_j$. Contractibility makes the inclusion $\{x_0\}\hookrightarrow A_j$ a homotopy equivalence, and thus $h^r(A_j,x_0)=0$ in every degree, by exactness and homotopy invariance of cohomology. The relative exact sequence for the triple $\{x_0\}\subseteq A_j\subseteq X$ givesConsequently each homogeneous $w_j$ lifts to $v_j\in h^{r_j}(X,A_j)$. Repeated relative multiplication givesNaturality maps this product to the product of the $w_j$ in $h^{\ast }(X,x_0)$. ThereforeFor nonhomogeneous classes the same conclusion follows by distributivity. This is nilpotence from a contractible subcomplex cover.

There is an alternative sufficient convention: if $X$ is path connected, the $A_j$ need not contain $x_0$. The absolute image $\bar{w}$ of a reduced class restricts to zero at $x_0$, hence at any other point since the point inclusions are homotopic along a path. On a nonempty contractible $A_j$ it therefore restricts to zero, so exactness lifts ${\bar{w}}_j$ to $h^{\ast }(X,A_j)$. Their relative product vanishes as above. The map $h^{\ast }(X,x_0)\to h^{\ast }(X)$ is injective: restriction to the basepoint is split surjective by the map $X\to \mathrm{pt}$, making the previous connecting map zero. Thus vanishing of the absolute product gives vanishing of the reduced product too.

Without either the pointed-subcomplex or connectedness convention, the printed conclusion is false. Take $X=\{x_0,x_1\}$ covered by its two singleton subcomplexes and $h^{\ast }=H^{\ast }(-;\mathbb{Z})$. Both subcomplexes are contractible, but $H^0(X,x_0)$ contains the function $w$ with values $0,1$. Pointwise multiplication gives $w^2=w\ne 0$. This states precisely the needed interpretation rather than assuming a contractible component lies in the basepoint component.

With the indicated points as basepoints, $X$ is the smash product ${\mathbb{CP}}^2\wedge {\mathbb{CP}}^2$. The quotient map identifies its positive-degree cohomology with the relative cohomology of the product modulo the wedge, which under the Künneth isomorphism is the ideal generated by $xy$. Thusin degrees $4,6,6,8$. The only nonzero product of positive-degree basis elements isTogether with the unit in degree zero, this determines the cohomology ring.