The following is exercise 5 on p. 176 in Hirsch's "Differential Topology" (corrected 6th printing):



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> Let $\eta = (p,E,B)$ be a fixed vector bundle over a compact manifold $B$, $\partial B = \varnothing$. An *$\eta$-submanifold* $(M,f)\subset V$ of a manifold $V$ is a pair $(M,f)$ where $M\subset V$ is a compact submanifold and $f$ is a bundle map from the normal bundle of $M$ to $\eta$ (this requires $\dim \eta = \dim V - \dim M$). Two $\eta$-submanifolds $(M\_i,f\_i)\subset V$ are $\eta$-*cobordant* if there is an $\eta$-submanifold $(W,f)\subset V\times I$ such that $\partial (W,f) = (M\_0,f\_0)\times 0 \cup (M\_1,f\_1)\times 1$ (using an obvious notation). The set of $\eta$-cobordism classes corresponds bijectively to the homotopy set $[V,E^\*]$.
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Notes: here $I = [0,1]$, $E^\*$ is the Thom space (or one-point compactification) of $E$, and "bundle map" means linear isomorphism on each fiber (see p. 88 of Hirsch).


**Questions**: are there references where this type of cobordism is introduced and/or studied in further detail? Is "$\eta$-cobordism" standard terminology for this type of cobordism?



A cobordism theory from Hirsch's "Differential Topology" (reference request)
As noted in the comments this is in Tom Diecks book (section 21.2), and in Wall's differential topology (Section 8.1).


