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For isometry families $f_x:U\to U\prime$ and $g_y:U\prime \to U\prime \prime$, the complement of $g_y{f}_x(V)$ decomposes orthogonally as the complement of $g_y(W)$ plus the image under $g_y$ of the complement of $f_x(V)$ in $W$. Their fiberwise one-point compactifications therefore give the same Thom space as the external direct sum. The resulting stage isomorphisms respect structure maps and give $Y{⋉}_g(X{⋉}_{f}E)\cong (Y\times X){⋉}_{g\circ f}E$ in the stable homotopy category.

A generalized cohomology theory with natural unital cup products compatible with relative groups, suspension, and the cohomology axioms. Its coefficient groups form a graded ring. Thom orientations make the cohomology of a bundle's Thom space a shifted module over the cohomology of the base.

// Target: geometry-and-topology.bigb

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

Let $\xi$ have real rank $r$, choose a bundle metric, and write $D(\xi ),S(\xi )$ for its disk and sphere bundles. In a multiplicative generalized cohomology theory, a Thom class in a generalized cohomology theory is a classwhose restriction to each fiber pair $(D^r,S^{r-1})$ is the suspension of the coefficient unit $1\in h^0(\mathrm{pt})$ under the chosen oriented identification. Equivalently, it is a fiberwise generator over the coefficient ring compatible with the $h$-orientation. Cup product with this class gives the Thom isomorphism theoremwhere $X$ is the base.

Let $z:X\to D(\xi )$ be the zero section and let $ȷ:h^r(D(\xi ),S(\xi ))\to h^r(D(\xi ))$ forget relative supports. The Euler class in a generalized cohomology theory isEquivalently, identify relative cohomology with reduced cohomology of the Thom space and pull its Thom class back along the zero section followed by the quotient map.

Suppose $s$ is a nowhere-zero section. Normalize it to $\hat{s}(x)=s(x)/\Vert s(x)\Vert \in S(\xi {)}_x$. The mapsgive a homotopy within $D(\xi )$ from $z$ to the sphere-valued section $\hat{s}$. The image $ȷ(u_h(\xi ))$ restricts to zero on $S(\xi )$, by the exact sequence of the pair. Homotopy invariance therefore yieldsThus a nowhere-zero section annihilates generalized Euler classes:The argument uses the actual nowhere-zero section and is valid for any chosen Thom orientation in the multiplicative theory.

For a real or complex vector bundle $E$, choose a fiber metric. Its Thom space is the pointed quotientcollapsing the sphere bundle to one basepoint. When the base is compact, this is the one-point compactification of the total space: fiberwise radial compression identifies the total space with the interior of $D(E)$, and all directions approaching its boundary become the single point at infinity.

Decompose ${\mathbb{C}}^{n+m+2}=V\oplus W$ with $V={\mathbb{C}}^{n+1}$, $W={\mathbb{C}}^{m+1}$, so the subspace being collapsed is $\mathbb{P}(V)={\mathbb{CP}}^n$. Every line outside it has a representative $(v,w)$ with $w\ne 0$. Its projection to $W$ gives the line $\ell =\mathbb{C}w\in \mathbb{P}(W)={\mathbb{CP}}^m$, and there is a unique complex-linear map $A:\ell \to V$ such that $A(w)=v$. The line in $V\oplus W$ is precisely the graph of $A$. This construction is unaffected by replacing $(v,w)$ with $(\lambda v,\lambda w)$.

Let $\gamma ={\eta }_1$ be the tautological line over $\mathbb{P}(W)$. The graph construction and its inverse are continuous, indeed smooth in the usual affine charts, and identifyA Hermitian metric gives a complex-linear bundle isomorphism $\bar{\gamma }\cong {\gamma }^{\ast }$. With the Hermitian form linear in its first argument, it sends $\bar{v}$ to the functional $w\mapsto ⟨w,v⟩$. Linearity in the conjugate vector is exactly what makes this a complex, rather than merely real, isomorphism. Thus the bundle is $({\bar{\eta }}_1{)}^{\oplus (n+1)}$, with the conjugation bars present in the original PDF.

Collapsing the closed subspace ${\mathbb{CP}}^n$ in compact Hausdorff space ${\mathbb{CP}}^{n+m+1}$ gives the one-point compactification of its open complement: neighborhoods of the collapsed point have compact complements, and conversely every compact subset of the complement is closed in the ambient space. The graph homeomorphism therefore extends over the added point. This proves the projective-space quotient as a Thom space of conjugate tautological lines:

The graph coordinates also determine the normal complex structures requested in the second part. The copy $\mathbb{P}(W)$ corresponds to the zero section of the graph bundle, whose normal bundle is that bundle itself. For a coordinate embedding ${\mathbb{CP}}^k\subset {\mathbb{CP}}^n$, use $W={\mathbb{C}}^{k+1}$ and its complementary coordinate space $V={\mathbb{C}}^{n-k}$. Then the normal bundle of a linear complex-projective embedding isThe complex structure is the linear structure in these graph coordinates. Equivalently, applying the preceding homeomorphism with its parameters $n-k-1$ and $k$ identifies the Thom space of this normal bundle with ${\mathbb{CP}}^n/{\mathbb{CP}}^{n-k-1}$, after reordering coordinates. Define ${\nu }_m$ in the identical way. The Hermitian-metric identification with the conjugate line does not change the complex normal orientation.

The original coordinate embeddings may be nested, so first move one into transverse position. Keep ${\mathbb{CP}}^k=\mathbb{P}(A)$ for the first $k+1$ coordinates, and move ${\mathbb{CP}}^m$ to $\mathbb{P}(B)$ for the last $m+1$ coordinates. A unitary coordinate permutation accomplishes this. Every unitary matrix can be joined to the identity by diagonalizing it and varying its eigenvalue phases continuously, so this move is an ambient isotopy. Its trace gives a cobordism carrying the transported normal complex structure, and hence leaves the input cobordism class unchanged.

If $k+m\ge n$, thenTheir projectivizations are transverse and meet in ${\mathbb{CP}}^r$, where $r=k+m-n$. The direct-sum normal structure from question 1 restricts toThe quotient map defining the transverse normal-bundle isomorphism is complex-linear here, so it agrees with the standard normal complex structure and contributes no orientation sign. If $k+m<n$, the first and last coordinate subspaces have no nonzero vector in common, so their projectivizations are disjoint.

Consequently the product of linear projective subspaces in complex cobordism isThe product degree is $2(2n-k-m)$, equal to the real codimension of the stated intersection when it exists. In the equality case $k+m=n$, the representative is one point with its complex-oriented normal space.

Write $M{U}^{\ast }={\Omega }_U^{\ast }$ for complex cobordism. A complex orientation of a smooth map is chosen stable complex normal data, not a complex structure on either manifold separately. More concretely, replace the continuous map by a homotopic smooth map $f_0$, choose an embeddingand give its normal bundle ${\nu }_j$ a stable complex structure on a real vector bundle. The auxiliary dimension can be chosen so the normal rank is even; further stabilization makes the normal bundle actually complex. Equivalence means stabilization by trivial complex summands and homotopy of the oriented embedding data. The stable identityidentifies this as orientation of the relative stable normal bundle. For odd relative codimension one uses an odd auxiliary suspension coordinate; the degree formula below is unchanged. This is the geometric factorization model for complex-oriented maps used in Algebraic and Geometric Topology 18, geometric cobordism discussion.

Let $d=\dim Y-\dim X$ and $r={\mathrm{rank}}_{\mathbb{R}}{\nu }_j=d+N$. The chosen normal orientation gives a Thom class $u_{{\nu }_j}$ of degree $r$ and the Thom isomorphism in complex cobordism. For compact manifolds the Pontryagin-Thom collapse of a tubular neighborhood to its Thom space isFor $a\in M{U}^q(X)$, define the Gysin map in complex cobordism by the compositeThus $f_{!}$ raises degree by the real relative codimension $d$. This definition depends on the specified orientation and is invariant under the stated equivalences of its normal data. The plus sign on a space denotes adjoining a disjoint basepoint.

For the embedding $i:L\hookrightarrow M$ with normal rank $2n$, let $c:M_{+}\to \mathrm{Th}(\nu )$ be its tubular collapse. The Thom construction gives $i_{!}(1)=c^{\ast }{u}_{\nu }$. Restricted to $L$, this collapse is exactly the zero section $z:L\to \mathrm{Th}(\nu )$. ThereforeThis is the self-intersection formula in complex cobordism. For an actual complex rank-$n$ normal bundle its top Chern class in complex cobordism is the zero-section pullback of its Thom class, so the right side is $c_n^{MU}(\nu )$. Equivalently, under the splitting principle for complex vector bundles both are the product of the first Chern classes of the line summands. The same argument gives $i^{\ast }{i}_{!}(a)=a{e}_{MU}(\nu )$ for every $a$.

There is a real distinction if the printed word “stable” is interpreted literally. The Euler class of a stably complex real bundle need not equal $c_n^{MU}$ of its stable complex class. For the diagonal embedding $S^4\hookrightarrow S^4\times S^4$, the normal bundle is $T{S}^4$. It has a stable complex structure withso its stable complex rank-two class is trivial and has second Chern class zero. However the ordinary integral image of $i^{\ast }{i}_{!}(1)$ is $e(T{S}^4)$, which evaluates to the Euler characteristic $2$. Thus the always-valid answer is the cobordism Euler class. The requested top-Chern identification is valid for an actual complex normal bundle, or if “top Chern class” here is explicitly being used as the Euler-class terminology for the chosen normal orientation.

A compact smooth embedding has a tubular neighborhood identified with its normal bundle. Collapse the complement of its open disk neighborhood to the basepoint and identify the remaining quotient with the Thom space of that normal bundle. Pulling back its Thom class constructs a Gysin class. For a graph embedding into a product with Euclidean space, desuspending the auxiliary coordinates gives the corresponding wrong-way map.

Decompose the ambient vector space as $V\oplus W$, of dimensions $n+1,m+1$. A projective line outside $\mathbb{P}(V)$ is the graph of a unique linear map from a line in $W$ to $V$. Thus the complement is the total space of ${\gamma }^{\ast }\otimes V$, which is complex-isomorphic to ${\bar{\gamma }}^{\oplus (n+1)}$ after choosing a Hermitian metric. Collapsing $\mathbb{P}(V)$ gives its one-point compactification, hence the Thom space because the base is compact.

// Target: geometry-and-topology.bigb

Given a compact parameter space $X$ and a family $f_x:U\to U\prime$ of linear isometries, a subordinate flag for a family of linear isometries consists of nested cofinal source and target subspaces $V_i,W_i$ with $f_x(V_i)\subset W_i$ for every $x$. After enlarging the target stages, compactness makes this possible for every finite source stage. It permits simultaneous finite-dimensional Thom space constructions.