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3 one more time

There are many spaces $X$ whose $K$ theory is trivial (isomorphic to that of a point), but whose ordinary cohomology is not. Famously, there is a 4-cell complex obtained by taking the homotopy cofiber of the "Adams map" $$f:S^{12}\cup_{3\iota} e^{13f:S^{11}\cup_{3\iota} e^{12} \to S^7\cup_{3\iota} e^8.$$ Here, $3\iota$ represents a degree $3$ self-map of a sphere.

The Adams map induces an isomorphism in $K$-theory, so $K^*(X)\approx K^*(*)\approx Z$. But $H^*(X,Z)\approx Z\oplus Z/3\oplus Z/3$.

(Hopefully I have the dimensions correct now.)

2 fixed dimensions again

There are many spaces $X$ whose $K$ theory is trivial (isomorphic to that of a point), but whose ordinary cohomology is not. Famously, there is a 4-cell complex obtained by taking the homotopy cofiber of the "Adams map" $$f:S^{10}\cup_{3\iota} e^{11f:S^{12}\cup_{3\iota} e^{13} \to S^6\cup_{3\iotaS^7\cup_{3\iota} e^7.$$ e^8.$$Here, 3\iota represents a degree 3 self-map of a sphere. The Adams map induces an isomorphism in K-theory, so K^*(X)\approx K^*(*)\approx Z. But H^*(X,Z)\approx Z\oplus Z/3\oplus Z/3. (Hopefully I have the dimensions correct now.) 1 There are many spaces X whose K theory is trivial (isomorphic to that of a point), but whose ordinary cohomology is not. Famously, there is a 4-cell complex obtained by taking the homotopy cofiber of the "Adams map"$$f:S^{10}\cup_{3\iota} e^{11} \to S^6\cup_{3\iota} e^7. Here, $3\iota$ represents a degree $3$ self-map of a sphere.

The Adams map induces an isomorphism in $K$-theory, so $K^*(X)\approx K^*(*)\approx Z$. But $H^*(X,Z)\approx Z\oplus Z/3\oplus Z/3$.