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830 boundaries are allowed to meet. |
830 boundaries are allowed to meet. |
831 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
831 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely |
832 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
832 the embeddings of a ``little" ball with image all of the big ball $B^n$. |
833 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
833 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?}) |
834 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
834 The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
835 (By shrinking the little balls (precomposing them with dilations), |
835 By shrinking the little balls (precomposing them with dilations), |
836 we see that both operads are homotopic to the space of $k$ framed points |
836 we see that both operads are homotopic to the space of $k$ framed points |
837 in $B^n$.) |
837 in $B^n$. |
838 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have the structure have |
838 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$ have |
839 an action of $\cE\cB_n$. |
839 an action of $\cE\cB_n$. |
840 \nn{add citation for this operad if we can find one} |
840 \nn{add citation for this operad if we can find one} |
841 |
841 |
842 \begin{example}[$E_n$ algebras] |
842 \begin{example}[$E_n$ algebras] |
843 \rm |
843 \rm |