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\begin{document}

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\section*{Connections, Curvature, and Cohomology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

This entry is about the book

\begin{itemize}%
\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]]

\emph{Connections, Curvature, and Cohomology}

Academic Press (1973)



\end{itemize}
on [[Chern-Weil theory]]: [[principal bundle]]s with [[connection on a bundle|connections]] and their [[characteristic class]]es.

Related books are

\begin{itemize}%
\item [[Theodore Frankel]], \emph{[[The Geometry of Physics - An Introduction]]}

\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{volume_i}{Volume I}\dotfill \pageref*{volume_i} \linebreak
\noindent\hyperlink{0_algebraic_and_analytic_preliminaries}{0 Algebraic and analytic preliminaries}\dotfill \pageref*{0_algebraic_and_analytic_preliminaries} \linebreak
\noindent\hyperlink{1_basic_concepts}{1 Basic concepts}\dotfill \pageref*{1_basic_concepts} \linebreak
\noindent\hyperlink{ii_vector_bundles}{II Vector bundles}\dotfill \pageref*{ii_vector_bundles} \linebreak
\noindent\hyperlink{iii_tangent_bundle_and_differential_forms}{III Tangent bundle and differential forms}\dotfill \pageref*{iii_tangent_bundle_and_differential_forms} \linebreak
\noindent\hyperlink{iv_calculus_of_differential_forms}{IV Calculus of differential forms}\dotfill \pageref*{iv_calculus_of_differential_forms} \linebreak
\noindent\hyperlink{v_de_rham_cohomology}{V De Rham cohomology}\dotfill \pageref*{v_de_rham_cohomology} \linebreak
\noindent\hyperlink{vi_mapping_degree}{VI Mapping degree}\dotfill \pageref*{vi_mapping_degree} \linebreak
\noindent\hyperlink{vii_integration_over_the_fiber}{VII Integration over the fiber}\dotfill \pageref*{vii_integration_over_the_fiber} \linebreak
\noindent\hyperlink{viii_cohomology_of_sphere_bundles}{VIII Cohomology of sphere bundles}\dotfill \pageref*{viii_cohomology_of_sphere_bundles} \linebreak
\noindent\hyperlink{ix_cohomology_of_vector_bundles}{IX Cohomology of vector bundles}\dotfill \pageref*{ix_cohomology_of_vector_bundles} \linebreak
\noindent\hyperlink{x_the_lefschetz_class_of_a_manifold}{X The Lefschetz class of a manifold}\dotfill \pageref*{x_the_lefschetz_class_of_a_manifold} \linebreak
\noindent\hyperlink{appendix_a_the_exponential_map}{Appendix A The exponential map}\dotfill \pageref*{appendix_a_the_exponential_map} \linebreak
\noindent\hyperlink{volume_ii}{Volume II}\dotfill \pageref*{volume_ii} \linebreak
\noindent\hyperlink{0_algebraic_and_analytic_preliminaries_2}{0 Algebraic and analytic preliminaries}\dotfill \pageref*{0_algebraic_and_analytic_preliminaries_2} \linebreak
\noindent\hyperlink{i_lie_groups}{I Lie groups}\dotfill \pageref*{i_lie_groups} \linebreak
\noindent\hyperlink{ii_subgroups_and_homogeneous_spaces}{II Subgroups and homogeneous spaces}\dotfill \pageref*{ii_subgroups_and_homogeneous_spaces} \linebreak
\noindent\hyperlink{iii_transformation_groups}{III Transformation groups}\dotfill \pageref*{iii_transformation_groups} \linebreak
\noindent\hyperlink{iv_invariant_cohomology}{IV Invariant cohomology}\dotfill \pageref*{iv_invariant_cohomology} \linebreak
\noindent\hyperlink{v_bundles_with_structrue_group}{V Bundles with structrue group}\dotfill \pageref*{v_bundles_with_structrue_group} \linebreak
\noindent\hyperlink{vi_principal_connections_and_the_weil_homomorphism}{VI Principal connections and the Weil homomorphism}\dotfill \pageref*{vi_principal_connections_and_the_weil_homomorphism} \linebreak
\noindent\hyperlink{vii_linear_connections}{VII Linear connections}\dotfill \pageref*{vii_linear_connections} \linebreak
\noindent\hyperlink{viii_characteristic_homomorphism_for_bundles}{VIII Characteristic homomorphism for $\Sigma$-bundles}\dotfill \pageref*{viii_characteristic_homomorphism_for_bundles} \linebreak
\noindent\hyperlink{ix_pontrjagin_pfaffian_chern_classes}{IX Pontrjagin, Pfaffian, Chern classes}\dotfill \pageref*{ix_pontrjagin_pfaffian_chern_classes} \linebreak
\noindent\hyperlink{x_the_gaussbonnetchern_theorem}{X The Gauss-Bonnet-Chern theorem}\dotfill \pageref*{x_the_gaussbonnetchern_theorem} \linebreak
\noindent\hyperlink{appendix_a_characteristic_coefficients_and_the_pfaffian}{Appendix A Characteristic coefficients and the Pfaffian}\dotfill \pageref*{appendix_a_characteristic_coefficients_and_the_pfaffian} \linebreak
\noindent\hyperlink{volume_iii}{Volume III}\dotfill \pageref*{volume_iii} \linebreak
\noindent\hyperlink{0_algebraic_preliminaries}{0 Algebraic preliminaries}\dotfill \pageref*{0_algebraic_preliminaries} \linebreak
\noindent\hyperlink{i_spectral_sequences}{I Spectral sequences}\dotfill \pageref*{i_spectral_sequences} \linebreak
\noindent\hyperlink{ii_koszul_complexes_of_spaces_and_algebras}{II Koszul complexes of $P$-spaces and $P$-algebras}\dotfill \pageref*{ii_koszul_complexes_of_spaces_and_algebras} \linebreak
\noindent\hyperlink{iii_koszul_complexes_of_differential_algebras}{III Koszul complexes of $P$-differential algebras}\dotfill \pageref*{iii_koszul_complexes_of_differential_algebras} \linebreak
\noindent\hyperlink{iv_lie_algebras_and_differential_spaces}{IV Lie algebras and differential spaces}\dotfill \pageref*{iv_lie_algebras_and_differential_spaces} \linebreak
\noindent\hyperlink{v_cohomology_of_lie_algebras_and_lie_groups}{V Cohomology of Lie algebras and Lie groups}\dotfill \pageref*{v_cohomology_of_lie_algebras_and_lie_groups} \linebreak
\noindent\hyperlink{vi_the_weil_alebra}{VI The Weil alebra}\dotfill \pageref*{vi_the_weil_alebra} \linebreak
\noindent\hyperlink{vii_operation_of_a_lie_algebra_in_a_graded_differential_algebra}{VII Operation of a Lie algebra in a graded differential algebra}\dotfill \pageref*{vii_operation_of_a_lie_algebra_in_a_graded_differential_algebra} \linebreak
\noindent\hyperlink{viii_algebraic_connections_and_principal_bundles}{VIII Algebraic connections and principal bundles}\dotfill \pageref*{viii_algebraic_connections_and_principal_bundles} \linebreak
\noindent\hyperlink{ix_cohomology_of_operations_and_principal_bundles}{IX Cohomology of operations and principal bundles}\dotfill \pageref*{ix_cohomology_of_operations_and_principal_bundles} \linebreak
\noindent\hyperlink{x_subalgebras}{X Subalgebras}\dotfill \pageref*{x_subalgebras} \linebreak
\noindent\hyperlink{xi_homogeneous_spaces}{XI Homogeneous spaces}\dotfill \pageref*{xi_homogeneous_spaces} \linebreak
\noindent\hyperlink{xii_operation_of_a_lie_algebra_on_a_pair}{XII Operation of a Lie algebra on a pair}\dotfill \pageref*{xii_operation_of_a_lie_algebra_on_a_pair} \linebreak
\noindent\hyperlink{appendix_a_characteristic_coefficients_and_the_pfaffian_2}{Appendix A Characteristic coefficients and the Pfaffian}\dotfill \pageref*{appendix_a_characteristic_coefficients_and_the_pfaffian_2} \linebreak


\hypertarget{volume_i}{}\subsection*{{Volume I}}\label{volume_i}

\hypertarget{0_algebraic_and_analytic_preliminaries}{}\subsubsection*{{0 Algebraic and analytic preliminaries}}\label{0_algebraic_and_analytic_preliminaries}

\begin{itemize}%
\item [[linear algebra]]


\item [[homological algebra]]


\item [[analysis]], [[topology]]



\end{itemize}
\hypertarget{1_basic_concepts}{}\subsubsection*{{1 Basic concepts}}\label{1_basic_concepts}

\begin{itemize}%
\item [[topological manifold]]


\item [[smooth manifold]]


\item [[fiber bundle]]



\end{itemize}
\hypertarget{ii_vector_bundles}{}\subsubsection*{{II Vector bundles}}\label{ii_vector_bundles}

\begin{itemize}%
\item [[vector bundle]]


\item [[section]]



\end{itemize}
\hypertarget{iii_tangent_bundle_and_differential_forms}{}\subsubsection*{{III Tangent bundle and differential forms}}\label{iii_tangent_bundle_and_differential_forms}

\begin{itemize}%
\item [[tangent bundle]]


\item [[vector field]]


\item [[differential form]]


\item [[orientation]]



\end{itemize}
\hypertarget{iv_calculus_of_differential_forms}{}\subsubsection*{{IV Calculus of differential forms}}\label{iv_calculus_of_differential_forms}

\begin{itemize}%
\item [[Cartan calculus]]


\item [[integration]]


\item [[Stokes theorem]]



\end{itemize}
\hypertarget{v_de_rham_cohomology}{}\subsubsection*{{V De Rham cohomology}}\label{v_de_rham_cohomology}

\begin{itemize}%
\item [[de Rham cohomology]]


\item [[Poincare duality]]


\item [[Künneth theorem]]


\item [[de Rham theorem]]



\end{itemize}
\hypertarget{vi_mapping_degree}{}\subsubsection*{{VI Mapping degree}}\label{vi_mapping_degree}

\hypertarget{vii_integration_over_the_fiber}{}\subsubsection*{{VII Integration over the fiber}}\label{vii_integration_over_the_fiber}

\begin{itemize}%
\item [[fiber integration]]

\end{itemize}
\hypertarget{viii_cohomology_of_sphere_bundles}{}\subsubsection*{{VIII Cohomology of sphere bundles}}\label{viii_cohomology_of_sphere_bundles}

\begin{itemize}%
\item [[Euler class]]

\end{itemize}
\hypertarget{ix_cohomology_of_vector_bundles}{}\subsubsection*{{IX Cohomology of vector bundles}}\label{ix_cohomology_of_vector_bundles}

\begin{itemize}%
\item [[Thom isomorphism]]

\end{itemize}
\hypertarget{x_the_lefschetz_class_of_a_manifold}{}\subsubsection*{{X The Lefschetz class of a manifold}}\label{x_the_lefschetz_class_of_a_manifold}

\begin{itemize}%
\item [[Lefschetz isomorphism]]

\end{itemize}
\hypertarget{appendix_a_the_exponential_map}{}\subsubsection*{{Appendix A The exponential map}}\label{appendix_a_the_exponential_map}

\hypertarget{volume_ii}{}\subsection*{{Volume II}}\label{volume_ii}

\hypertarget{0_algebraic_and_analytic_preliminaries_2}{}\subsubsection*{{0 Algebraic and analytic preliminaries}}\label{0_algebraic_and_analytic_preliminaries_2}

\hypertarget{i_lie_groups}{}\subsubsection*{{I Lie groups}}\label{i_lie_groups}

\begin{itemize}%
\item [[Lie group]]

\end{itemize}
\hypertarget{ii_subgroups_and_homogeneous_spaces}{}\subsubsection*{{II Subgroups and homogeneous spaces}}\label{ii_subgroups_and_homogeneous_spaces}

\begin{itemize}%
\item [[homogeneous space]]

\end{itemize}
\hypertarget{iii_transformation_groups}{}\subsubsection*{{III Transformation groups}}\label{iii_transformation_groups}

\hypertarget{iv_invariant_cohomology}{}\subsubsection*{{IV Invariant cohomology}}\label{iv_invariant_cohomology}

\begin{itemize}%
\item [[equivariant cohomology]]

\end{itemize}
\hypertarget{v_bundles_with_structrue_group}{}\subsubsection*{{V Bundles with structrue group}}\label{v_bundles_with_structrue_group}

\begin{itemize}%
\item [[principal bundle]]

\end{itemize}
\hypertarget{vi_principal_connections_and_the_weil_homomorphism}{}\subsubsection*{{VI Principal connections and the Weil homomorphism}}\label{vi_principal_connections_and_the_weil_homomorphism}

\begin{itemize}%
\item [[connection on a bundle]]


\item [[Chern-Weil homomorphism]]



\end{itemize}
\hypertarget{vii_linear_connections}{}\subsubsection*{{VII Linear connections}}\label{vii_linear_connections}

\hypertarget{viii_characteristic_homomorphism_for_bundles}{}\subsubsection*{{VIII Characteristic homomorphism for $\Sigma$-bundles}}\label{viii_characteristic_homomorphism_for_bundles}

\hypertarget{ix_pontrjagin_pfaffian_chern_classes}{}\subsubsection*{{IX Pontrjagin, Pfaffian, Chern classes}}\label{ix_pontrjagin_pfaffian_chern_classes}

\begin{itemize}%
\item [[Pontryagin class]]


\item [[Pfaffian]]


\item [[Chern class]]



\end{itemize}
\hypertarget{x_the_gaussbonnetchern_theorem}{}\subsubsection*{{X The Gauss-Bonnet-Chern theorem}}\label{x_the_gaussbonnetchern_theorem}

\begin{itemize}%
\item [[Gauss-Bonnet theorem]]

\end{itemize}
\hypertarget{appendix_a_characteristic_coefficients_and_the_pfaffian}{}\subsubsection*{{Appendix A Characteristic coefficients and the Pfaffian}}\label{appendix_a_characteristic_coefficients_and_the_pfaffian}

\hypertarget{volume_iii}{}\subsection*{{Volume III}}\label{volume_iii}

\hypertarget{0_algebraic_preliminaries}{}\subsubsection*{{0 Algebraic preliminaries}}\label{0_algebraic_preliminaries}

\hypertarget{i_spectral_sequences}{}\subsubsection*{{I Spectral sequences}}\label{i_spectral_sequences}

\begin{itemize}%
\item [[fiber sequence]]


\item [[spectral sequence]]



\end{itemize}
\hypertarget{ii_koszul_complexes_of_spaces_and_algebras}{}\subsubsection*{{II Koszul complexes of $P$-spaces and $P$-algebras}}\label{ii_koszul_complexes_of_spaces_and_algebras}

\hypertarget{iii_koszul_complexes_of_differential_algebras}{}\subsubsection*{{III Koszul complexes of $P$-differential algebras}}\label{iii_koszul_complexes_of_differential_algebras}

\hypertarget{iv_lie_algebras_and_differential_spaces}{}\subsubsection*{{IV Lie algebras and differential spaces}}\label{iv_lie_algebras_and_differential_spaces}

\begin{itemize}%
\item [[Lie algebra]]

\end{itemize}
\hypertarget{v_cohomology_of_lie_algebras_and_lie_groups}{}\subsubsection*{{V Cohomology of Lie algebras and Lie groups}}\label{v_cohomology_of_lie_algebras_and_lie_groups}

\begin{itemize}%
\item [[Lie algebra cohomology]]


\item [[Lie group cohomology]]



\end{itemize}
\hypertarget{vi_the_weil_alebra}{}\subsubsection*{{VI The Weil alebra}}\label{vi_the_weil_alebra}

\begin{itemize}%
\item [[Weil algebra]]

\end{itemize}
\hypertarget{vii_operation_of_a_lie_algebra_in_a_graded_differential_algebra}{}\subsubsection*{{VII Operation of a Lie algebra in a graded differential algebra}}\label{vii_operation_of_a_lie_algebra_in_a_graded_differential_algebra}

\hypertarget{viii_algebraic_connections_and_principal_bundles}{}\subsubsection*{{VIII Algebraic connections and principal bundles}}\label{viii_algebraic_connections_and_principal_bundles}

\hypertarget{ix_cohomology_of_operations_and_principal_bundles}{}\subsubsection*{{IX Cohomology of operations and principal bundles}}\label{ix_cohomology_of_operations_and_principal_bundles}

\hypertarget{x_subalgebras}{}\subsubsection*{{X Subalgebras}}\label{x_subalgebras}

\hypertarget{xi_homogeneous_spaces}{}\subsubsection*{{XI Homogeneous spaces}}\label{xi_homogeneous_spaces}

\hypertarget{xii_operation_of_a_lie_algebra_on_a_pair}{}\subsubsection*{{XII Operation of a Lie algebra on a pair}}\label{xii_operation_of_a_lie_algebra_on_a_pair}

\hypertarget{appendix_a_characteristic_coefficients_and_the_pfaffian_2}{}\subsubsection*{{Appendix A Characteristic coefficients and the Pfaffian}}\label{appendix_a_characteristic_coefficients_and_the_pfaffian_2}

category: reference



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